9.1E: Introduction to Linear Higher Order Equations (Exercises) - Mathematics


1. Verify that the given function is the solution of the initial value problem.

  1. (x^3y'''-3x^2y''+6xy'-6y=dfrac{-24}{ x}, quad y(-1)=0), (y'(-1)=0, quad y''(-1)=0) ;(y=-6x-8x^2-3x^3 + {1over x})
  2. (y'''- dfrac{1}{x}y''-y'+ dfrac{1}{x}y= dfrac{x^2-4}{x^4}, quad y(1)= dfrac{3}{2}, quad y'(1)= dfrac{1}{2}, y''(1)=1) ;(y=x+ dfrac{1}{2x})
  3. (xy'''-y''-xy'+y=x^2, quad y(1)=2,quad y'(1)=5,quad y''(1)=-1) ;(y=-x^2-2+2e^{(x-1)}-e^{-(x-1)}+4x)
  4. (4x^3y'''+4x^2y''-5xy'+2y=30x^2, quad y(1)=5,quad y'(1)= dfrac{17}{2}) ;(y''(1)= dfrac{63}{4};quad y=2x^2ln x-x^{1/2}+2x^{-1/2}+4x^2)
  5. (x^4y^{(4)}-4x^3y'''+12x^2y''-24xy'+24y=6x^4, quad y(1)=-2) ;(y'(1)=-9, quad y''(1)=-27,quad y'''(1)=-52) ;(y=x^4ln x+x-2x^2+3x^3-4x^4)
  6. (xy^{(4)}-y'''-4xy''+4y'=96x^2, quad y(1)=-5,quad y'(1)=-24) ;(y''(1)=-36; quad y'''(1)=-48;quad y=9-12x+6x^2-8x^3)

2. Solve the initial value problem

[x^3y'''-x^2 y''-2xy'+6y=0, quad y(-1)=-4, quad y'(-1)=-14,quad y''(-1)=-20. onumber ] HINT: See Example 9.1.1.

3. Solve the initial value problem

[y^{(4)}+y'''-7y''-y'+6y=0, quad y(0)=5,quad y'(0)=-6,quad y''(0)=10,quad y'''(0)-36. onumber ] HINT: See Example 9.1.2.

4. Find solutions (y_1), (y_2), …, (y_n) of the equation (y^{(n)}=0) that satisfy the initial conditions

[y_i^{(j)}(x_0)=left{egin{array}{cl} 0,&j e i-1,[5 pt] 1,&j=i-1,end{array} ight.; 1le ile n. onumber ]


  1. Verify that the function [y=c_1x^3+c_2x^2+{c_3over x} onumber ] satisfies [x^3 y'''-x^2y''-2xy'+6y=0 ag{A}] if (c_1), (c_2), and (c_3) are constants.
  2. Use (a) to find solutions (y_1), (y_2), and (y_3) of (A) such that [egin{array}{rl} y_1(1)&=1,quad y_1'(1)=0,quad y_1''(1)=0 [5 pt] y_2(1)&=0,quad y_2'(1)=1,quad y_2''(1)=0 [5 pt] y_3(1)&=0,quad y_3'(1)=0,quad y_3''(1)=1. end{array} onumber ]
  3. Use (b) to find the solution of (A) such that [y(1)=k_0,quad y'(1)=k_1,quad y''(1)=k_2. onumber ]

6. Verify that the given functions are solutions of the given equation, and show that they form a fundamental set of solutions of the equation on any interval on which the equation is normal.

  1. (y'''+y''-y'-y=0; quad{e^x,,e^{-x},,xe^{-x}})
  2. (y'''-3y''+7y'-5y=0; quad{e^x,,e^xcos2x,,e^xsin2x}).
  3. (xy'''-y''-xy'+y=0; quad {e^x,,e^{-x},,x})
  4. (x^2y'''+2xy''-(x^2+2)y=0; quad {e^x/ x,,e^{-x}/ x,,1})
  5. ((x^2-2x+2)y'''-x^2y''+2xy'-2y=0; quad {x,,x^2,,e^x} )
  6. ((2x-1)y^{(4)}-4xy'''+(5-2x)y''+4xy'-4y=0; quad{x,,e^x,,e^{-x},e^{2x}})
  7. (xy^{(4)}-y'''-4xy'+4y'=0; quad{1,x^2,,e^{2x},,e^{-2x}})

7. Find the Wronskian (W) of a set of three solutions of [y'''+2xy''+e^xy'-y=0, onumber ] given that (W(0)=2).

8. Find the Wronskian (W) of a set of four solutions of [y^{(4)}+( an x)y'''+x^2y''+2xy=0, onumber ] given that (W(pi/4)=K).


  1. Evaluate the Wronskian (W) ({e^x,,xe^x,, x^2e^x}). Evaluate (W(0)).
  2. Verify that (y_1), (y_2), and (y_3) satisfy [y'''-3y''+3y'-y=0. ag{A}]
  3. Use (W(0)) from (a) and Abel’s formula to calculate (W(x)).
  4. What is the general solution of (A)?

10. Compute the Wronskian of the given set of functions.

  1. ({1,,e^x,,e^{-x}})
  2. ({e^x,, e^xsin x,,e^xcos x})
  3. ({2,,x+1,,x^2+2})
  4. (x,,xln x,,1/x})
  5. ({1,,x,,{x^2over2!},, {x^3over3!},,cdots,,{x^nover n!}})
  6. ({e^x,,e^{-x},,x})
  7. ({e^x/x,,e^{-x}/x,,1})
  8. ({x,,x^2,,e^x})
  9. ({x,,x^3,,1/x,,1/x^2})
  10. ({e^x,,e^{-x},,x,,e^{2x}})
  11. ({e^{2x},,e^{-2x},,1,,x^2})

11. Suppose (Ly=0) is normal on ((a,b)) and (x_0) is in ((a,b)). Use Theorem 9.1.1 to show that (yequiv0) is the only solution of the initial value problem [Ly=0, quad y(x_0)=0,quad y'(x_0)=0,dots, y^{(n-1)}(x_0)=0, onumber ] on ((a,b)).

12. Prove: If (y_1), (y_2), …, (y_n) are solutions of (Ly=0) and the functions [z_i=sum^n_{j=1}a_{ij}y_j,quad 1le ile n, onumber ] form a fundamental set of solutions of (Ly=0), then so do (y_1), (y_2), …, (y_n).

13. Prove: If [y=c_1y_1+c_2y_2+cdots+c_ky_k+y_p onumber ] is a solution of a linear equation (Ly=F) for every choice of the constants (c_1), (c_2),…, (c_k), then (Ly_i=0) for (1le ile k).

14. Suppose (Ly=0) is normal on ((a,b)) and let (x_0) be in ((a,b)). For (1le ile n), let (y_i) be the solution of the initial value problem [Ly_i=0, quad y_i^{(j)} (x_0)= left{egin{array}{cl} 0,& j e i-1, 1,&j=i-1,end{array} ight. 1le ile n, onumber ] where (x_0) is an arbitrary point in ((a,b)). Show that any solution of (Ly=0) on ((a, b)), can be written as [y=c_1y_1+c_2y_2+cdots+c_ny_n, onumber ] with (c_j=y^{(j-1)}(x_0)).

15. Suppose ({y_1, y_2,dots, y_n}) is a fundamental set of solutions of [ P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+cdots+P_n(x)y=0 onumber ] on ((a,b)), and let [egin{array}{rl} z_1&=a_{11}y_1+a_{12}y_2+cdots+a_{1n}y_n z_2&=a_{21}y_1+a_{22}y_2+cdots+a_{2n}y_n phantom{z_1}&vdotsphantom{_1y_1+a}vdots phantom{_2y_2+cdots+a}vdotsphantom{_ny_n} phantom{=b}vdots z_n&=a_{n1}y_1+a_{n2}y_2+cdots+a_{nn}y_n, end{array} onumber ] where the ({a_{ij}}) are constants. Show that ({z_1, z_2,dots, z_n}) is a fundamental set of solutions of (A) if and only if the determinant [left|egin{array}{cccc} a_{11}&a_{12}&cdots&a_{1n} a_{21}&a_{22}&cdots&a_{2n} vdots&vdots&ddots&vdots a_{n1}&a_{n2}&cdots&a_{nn}end{array} ight| onumber ] is nonzero. HINT: The determinant of a product of (n imes n) matrices equals the product of the determinants.

16. Show that ({y_1,y_2,dots,y_n}) is linearly dependent on ((a,b)) if and only if at least one of the functions (y_1), (y_2), …, (y_n) can be written as a linear combination of the others on ((a,b)).


Take the following as a hint in Exercises 9.1.17-9.1.19:

By the definition of determinant, [left|egin{array}{cccc}{a_{11}}&{a_{12}}&{cdots }&{a_{1n}}{a_{21}}&{a_{22}}&{cdots }&{a_{2n}}{vdots }&{vdots }&{ddots }&{vdots }{a_{n1}}&{a_{n2}}&{cdots }&{a_{nn}} end{array} ight| = sumpm a_{1i_{1}}a_{2i_{2}},cdots , a_{ni_{n}}, onumber ] where the sum is over all permutations ((i_{i}, i_{2}, cdots , i_{n})) of ((1,2,cdots ,n)) and the choice of (+) or (-) in each term depends only on the permutation associated with that term.

17. Prove: If [A(u_1,u_2,dots,u_n)= left|egin{array}{cccc} a_{11}&a_{12}&cdots&a_{1n}[4pt] a_{21}&a_{22}&cdots&a_{2n}[4pt] vdots&vdots&ddots&vdots[4pt] a_{n-1,1}&a_{n-1,2}&cdots&a_{n-1,n}[4pt] u_1&u_2&cdots&u_nend{array} ight|, onumber ] then [A(u_1+v_1, u_2+v_2,dots, u_n+v_n)=A(u_1,u_2,dots,u_n)+A(v_1,v_2,dots, v_n). onumber ]

18. Let [F=left|egin{array}{cccc} f_{11}&f_{12}&cdots&f_{1n}[4pt] f_{21}&f_{22}&cdots&f_{2n}[4pt] vdots&vdots&ddots&vdots[4pt] f_{n1}&f_{n2}&cdots&f_{nn}end{array} ight|, onumber ] where (f_{ij}; (1le i,; jle n)) is differentiable. Show that [F'=F_1+F_2+cdots+F_n, onumber ] where (F_i) is the determinant obtained by differentiating the (i)th row of (F).

19. Use Exercise 9.1.18 to show that if (W) is the Wronskian of the (n)-times differentiable functions (y_1), (y_2), …, (y_n), then

[W'= left|egin{array}{cccc} y_1&y_2&cdots&y_n[4pt] y'_1&y'_2&cdots&y'_n[4pt] vdots&vdots&ddots&vdots[4pt] y_1^{(n-2)}&y_2^{(n-2)}&cdots&y_n^{(n-2)}[4pt] y_1^{(n)}&y_2^{(n)}&cdots&y_n^{(n)} end{array} ight|. onumber ]


20. Use Exercises 9.1.17 and 9.1.19 to show that if (W) is the Wronskian of solutions ({y_1,y_2,dots,y_n}) of the normal equation [P_0(x)y^{(n)}+P_1(x)y^{(n-1)}+cdots+P_n(x)y=0, ag{A}] then (W'=-P_1W/P_0). Derive Abel’s formula (Equation 9.1.15) from this.

21. Prove Theorem 9.1.6.

22. Prove Theorem 9.1.7.

23. Show that if the Wronskian of the (n)-times continuously differentiable functions ({y_1,y_2,dots,y_n}) has no zeros in ((a,b)), then the differential equation obtained by expanding the determinant [left|egin{array}{ccccc} y&y_1&y_2&cdots&y_n[4pt] y'&y'_1&y'_2&cdots&y'_n[4pt] vdots&vdots&vdots&ddots& vdots[4pt] y^{(n)}&y_{1}^{(n)}&y_2^{(n)}&cdots&y_n^{(n)} end{array} ight|=0, onumber ] in cofactors of its first column is normal and has ({y_1,y_2,dots,y_n}) as a fundamental set of solutions on ((a,b)).

24. Use the method suggested by Exercise 9.1.23 to find a linear homogeneous equation such that the given set of functions is a fundamental set of solutions on intervals on which the Wronskian of the set has no zeros.

  1. ({x,,x^2-1,,x^2+1})
  2. ({e^x,,e^{-x},,x})
  3. ({e^x,,xe^{-x},,1})
  4. ({x,,x^2,,e^x})
  5. ({x,,x^2,,1/x})
  6. ({x+1,,e^x,,e^{3x}})
  7. ({x,,x^3,,1/x,,1/x^2})
  8. ({x,,xln x,,1/x,,x^2})
  9. ({e^x,,e^{-x},,x,,e^{2x}})
  10. ({e^{2x},,e^{-2x},,1,,x^2})

Differential Equations : Linear Equations

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

We then solve the characteristic equation and find that (Use the quadratic formula if you'd like) This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, . Plugging in our initial condition, we find that . To plug in the second initial condition, we take the derivative and find that . Plugging in the second initial condition yields . Solving this simple system of linear equations shows us that

Leaving us with a final answer of

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

Example Question #1 : Higher Order Differential Equations

Find the general solution to .

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

To factor this, in this case we may use factoring by grouping. More generally, we may use horner's scheme/synthetic division to test possible roots. Here are both methods shown.

Alternatively, the rational root theorem suggests that we try -1 or 1 as a root of this equation. Using horner's scheme, we see

Which tells us the the polynomial factors into and that . This means that the fundamental set of solutions is

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, . As this is not an initial value problem and just asks for the general solution, we are done.

Example Question #1 : Higher Order Differential Equations

Solve the initial value problem for and .

This is a linear higher order differential equation. First, we need the characteristic equation, which is just obtained by turning the derivative orders into powers to get the following:

We then solve the characteristic equation and find that This lets us know that the basis for the fundamental set of solutions to this problem (solutions to the homogeneous problem) contains .

As the given problem was homogeneous, the solution is just a linear combination of these functions. Thus, . Plugging in our initial condition, we find that . To plug in the second initial condition, we take the derivative and find that . Plugging in the second initial condition yields . Solving this simple system of linear equations shows us that

Leaving us with a final answer of

(Note, it would have been very simple to find the right answer just by taking derivatives and plugging in, but this is not overly helpful for non-multiple choice questions)

Example Question #1 : Higher Order Differential Equations

Solve the following homogeneous differential equation:

The ode has a characteristic equation of .

This yields the double root of r=2. Then the roots are plugged into the general solution to a homogeneous differential equation with a repeated root.

Example Question #1 : Higher Order Differential Equations

Solve the General form of the differential equation:

Where and are arbitrary constants

Where and are arbitrary constants

Where and are arbitrary constants

Where and are arbitrary constants

Where and are arbitrary constants

This differential equation has a characteristic equation of

, which yields the roots for r=2 and r=3. Once the roots or established to be real and non-repeated, the general solution for homogeneous linear ODEs is used. this equation is given as:

with r being the roots of the characteristic equation.

Example Question #1 : Higher Order Differential Equations

Solve the general homogeneous part of the following differential equation:

Where and are arbitrary but not meaningless constants

Where and are arbitrary but not meaningless constants

Where and are arbitrary but not meaningless constants

Where and are arbitrary but not meaningless constants

Where and are arbitrary but not meaningless constants

We start off by noting that the homogeneous equation we are trying to solve is given as

This differential equation thus has characteristic equation of

This has roots of r=3 and r=-4, therefore, the general homogeneous solution is given by:

Example Question #1 : Higher Order Differential Equations

Solve the following homogeneous differential equation:

This differential equation has characteristic equation of:

It must be noted that this characteristic equation has a double root of r=5.

Thus the general solution to a homogeneous differential equation with a repeated root is used.

in the case of a repeated root such as this, and is the repeated root r=5.

Therefore, the solution is

Example Question #1 : Higher Order Differential Equations

Find a general solution to the following Differential Equation

Solving the auxiliary equation

Trying out candidates for roots from the Rational Root Theorem we have a root .

Factoring completely we have

where are arbitrary constants.

All Differential Equations Resources

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Book Description

The book takes a problem solving approach in presenting the topic of differential equations. It provides a complete narrative of differential equations showing the theoretical aspects of the problem (the how's and why's), various steps in arriving at solutions, multiple ways of obtaining solutions and comparison of solutions. A large number of comprehensive examples are provided to show depth and breadth and these are presented in a manner very similar to the instructor's class room work. The examples contain solutions from Laplace transform based approaches alongside the solutions based on eigenvalues and eigenvectors and characteristic equations. The verification of the results in examples is additionally provided using Runge-Kutta offering a holistic means to interpret and understand the solutions. Wherever necessary, phase plots are provided to support the analytical results. All the examples are worked out using MATLAB® taking advantage of the Symbolic Toolbox and LaTex for displaying equations. With the subject matter being presented through these descriptive examples, students will find it easy to grasp the concepts. A large number of exercises have been provided in each chapter to allow instructors and students to explore various aspects of differential equations.

MATH 117. Precalculus I

Required text(s): Eric Connally, Hughes-Hallett, D., and Gleason, A. M. Functions Modeling Change: A Preparation for Calculus. 5th ed. New Jersey: Wiley, 2015. Packaged with WileyPlus.​

Prerequisites: MATH 100 or Math Diagnostic Test

Course description: Inverse functions, quadratic functions, complex numbers. Detailed study of polynomial functions including zeros, factor theorem, and graphs. Rational functions, exponential and logarithmic functions and their applications. Systems of equations, inequalities, partial fractions, linear programming, sequences and series. Word problems are emphasized throughout the course.

Mathematical Sciences

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Problem solving is a fundamental mathematical skill. In this course students will be exposed to problems coming from a wide range of mathematical disciplines and will work together in a collaborative environment to explore potential solutions. Discussion problems may be inspired by the research of faculty leading the discussion, by past mathematical competitions (such as the Putnam Competition) or elsewhere. This course meets once per week, with an emphasis on discussion and exploration of problems. There will be no exam and no assigned homework. Grading is by participation only. This course may be taken multiple times content will vary depending on the speakers. Grading for this course will be on a Pass/NR basis. Recommended background: Curiosity about Mathematics


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Cat. II This course introduces the concepts and techniques of combinatorics, a part of mathematics with applications in computer science and in the social, biological, and physical sciences. Emphasis will be given to problem solving. Topics will be selected from: basic counting methods, inclusion-exclusion principle, generating functions, recurrence relations, systems of distinct representatives, combinatorial designs, combinatorial algorithms and applications of combinatorics. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying combinatorics. Undergraduate credit may not be earned both for this course and for MA 3273. Recommended background: MA 2071. This course will be offered in 2015-16, and in alternating years thereafter.


Cat. I This course focuses on the principles of building mathematical models from a physical, chemical or biological system and interpreting the results. Students will learn how to construct a mathematical model and will be able to interpret solutions of this model in terms of the context of the application. Mathematical topics focus on solving systems of ordinary differential equations, and may include the use of stability theory and phase-plane analysis. Applications will be chosen from electrical and mechanical oscillations, control theory, ecological or epidemiological models and reaction kinetics. This course is designed primarily for students interested in the deeper mathematical issues underlying mathematical modeling. Students may be required to use programming languages such as Matlab or Maple to further investigate different models. Recommended background: multivariable calculus (MA 1024 or equivalent), ordinary differential equations (MA 2051 or equivalent), and linear algebra (MA 2071 or equivalent).


Cat. I This course is designed to introduce the student to statistical methods and concepts commonly used in the life sciences. Emphasis will be on the practical aspects of statistical design and analysis with examples drawn exclusively from the life sciences, and students will collect and analyze data. Topics covered include analytic and graphical and numerical summary measures, probability models for sampling distributions, the central limit theorem, and one and two sample point and interval estimation, parametric and non-parametric hypothesis testing, principles of experimental design, comparisons of paired samples and categorical data analysis. Undergraduate credit may not be earned for both this course and for MA 2611. Recommended background: MA 1022.


Cat. I This course is designed to introduce the student to data analytic and applied statistical methods commonly used in industrial and scientific applications as well as in course and project work at WPI. Emphasis will be on the practical aspects of statistics with students analyzing real data sets on an interactive computer package. Topics covered include analytic and graphical representation of data, exploratory data analysis, basic issues in the design and conduct of experimental and observational studies, the central limit theorem, one and two sample point and interval estimation and tests of hypotheses. Recommended background: MA 1022.


Cat. I This course is a continuation of MA 2611. Topics covered include simple and multiple regression, one and two-way tables for categorical data, design and analysis of one factor experiments and distribution-free methods. Recommended background: MA 2611.


Cat. I This course is designed to introduce the student to probability. Topics to be covered are: basic probability theory including Bayes theorem discrete and continuous random variables special distributions including the Bernoulli, Binomial, Geometric, Poisson, Uniform, Normal, Exponential, Chisquare, Gamma, Weibull, and Beta distributions multivariate distributions conditional and marginal distributions independence expectation transformations of univariate random variables. Recommended background: MA 1024.


Cat. I The purpose of this course is twofold: - To introduce the student to probability. Topics to be covered will be chosen from: axiomatic development of probability independence Bayes theorem discrete and continuous random variables expectation special distributions including the binomial and normal moment generating functions multivariate distributions conditional and marginal distributions independence of random variables transformations of random variables limit theorems. - To introduce fundamental ideas and methods of mathematics using the study of probability as the vehicle. These ideas and methods may include systematic theorem-proof development starting with basic axioms mathematical induction set theory applications of univariate and multivariate calculus. This course is designed primarily for Mathematical Sciences majors and those interested in the deeper mathematical issues underlying probability theory. Recommended background: MA 1024. Undergraduate credit may not be earned both for this course and for MA 2621.





A study of actuarial mathematics with emphasis on the theory and application of contingency mathematics in various areas of insurance. Topics usually included are: survival functions and life tables life insurance property insurance annuities net premiums and premium reserves. Recommended background: An introduction to the theory of interest, and familiarity with basic probability (MA 2211 and either MA 2621 or MA 2631, or equivalent).


A continuation of the study of actuarial mathematics with emphasis on calculations in various areas of insurance, based on multiple insureds, multiple decrements, and multiple state models. Topics usually included are: survival functions life insurance property insurance common shock Poisson processes and their application to insurance settings gross premiums and reserves. Recommended background: An introduction to actuarial mathematics (MA 3212 or equivalent)


Cat. I The mathematical subject of linear programming deals with those problems in optimal resource allocation which can be modeled by a linear profit (or cost) function together with feasibility constraints expressible as linear inequalities. Such problems arise regularly in many industries, ranging from manufacturing to transportation, from the design of livestock diets to the construction of investment portfolios. This course considers the formulation of such real-world optimization problems as linear programming problems, the most important algorithms for their solution, and techniques for their analysis. The core material includes problem formulation, the primal and dual simplex algorithms, and duality theory. Further topics may include: sensitivity analysis applications such as matrix games or network flow models bounded variable linear programs interior point methods. Recommended background: Matrices and Linear Algebra (MA 2071, or equivalent).


Cat. II Discrete optimization is a lively field of applied mathematics in which techniques from combinatorics, linear programming, and the theory of algorithms are used to solve optimization problems over discrete structures, such as networks or graphs. The course will emphasize algorithmic solutions to general problems, their complexity, and their application to real-world problems drawn from such areas as VLSI design, telecommunications, airline crew scheduling, and product distribution. Topics will be selected from: Network flow, optimal matching, integrality of polyhedra, matroids, and NP-completeness. Recommended background: At least one course in graph theory, combinatorics or optimization (e.g., MA 2271, MA 2273 or MA 3231).


Cat. I This course provides an introduction to modern computational methods for linear and nonlinear equations and systems and their applications. Topics covered include: solution of nonlinear scalar equations, direct and iterative algorithms for the solution of systems of linear equations, solution of nonlinear systems, the eigenvalue problem for matrices. Error analysis will be emphasized throughout. Recommended background: MA 2071. An ability to write computer programs in a scientific language is assumed.


Cat. I This course provides an introduction to modern computational methods for differential and integral calculus and differential equations. Topics covered include: interpolation and polynomial approximation, approximation theory, numerical differentiation and integration, numerical solutions of ordinary differential equations. Error analysis will be emphasized throughout. Recommended background: MA 2051. An ability to write computer programs in a scientific language is assumed. Undergraduate credit may not be earned for both this course and for MA 3255/CS 4031.


Cat. II The first part of the course will cover existence and uniqueness of solutions, continuous dependence of solutions on parameters and initial conditions, maximal interval of existence of solutions, Gronwall's inequality, linear systems and the variation of constants formula, Floquet theory, stability of linear and perturbed linear systems. The second part of the course will cover material selected by the instructor. Possible topics include: Introduction to dynamical systems, stability by Lyapunov's direct method, study of periodic solutions, singular perturbation theory and nonlinear oscillation theory. Recommended background: MA 2431 and MA 3832. This course will be offered in 2015-16, and in alternating years thereafter.


Cat. II This course covers the calculus of variations and select topics from optimal control theory. The purpose of the course is to expose students to mathematical concepts and techniques needed to handle various problems of design encountered in many fields, e. g. electrical engineering, structural mechanics and manufacturing. Topics covered will include: derivation of the necessary conditions of a minimum for simple variational problems and problems with constraints, variational principles of mechanics and physics, direct methods of minimization of functions, Pontryagin's maximum principle in the theory of optimal control and elements of dynamic programming. Recommended background: MA 2051. This course will be offered in 2016-17, and in alternating years thereafter.


Cat. II This course will teach students how to design experiments in order to collect meaningful data for analysis and decision making. This course continues the exploration of statistics for scientific and industrial applications begun in MA 2611 and MA 2612. The course offers comprehensive coverage of the key elements of experimental design used by applied researchers to solve problems in the field, such as random assignment, replication, blocking, and confounding. Topics covered include the design and analysis of general factorial experiments two-level factorial and fractional factorial experiments principles of design completely randomized designs and one-way analysis of variance (ANOVA) complete block designs and two-way analysis of variance complete factorial experiments fixed, random, and mixed models split-plot designs nested designs. Recommended background: Applied Statistics (MA 2611 and MA2612, or equivalent).


Cat. I This course introduces students to the mathematical principles of statistics. Topics will be chosen from: Sampling distributions, limit theorems, point and interval estimation, sufficiency, completeness, efficiency, consistency the Rao- Blackwell theorem and the Cramer-Rao bound minimum variance unbiased estimators and maximum likelihood estimators tests of hypotheses including the Neyman-Pearson lemma, uniformly most powerful and likelihood radio tests. Recommended background: MA 2631.


This course provides an introduction to one of the major areas of modern algebra. Topics covered include: groups, subgroups, permutation groups, normal subgroups, factor groups, homomorphisms, isomorphisms and the fundamental homomorphism theorem. Recommended background: MA 2073.


Cat. II This course provides an introduction to one of the major areas of modern algebra. Topics covered include: rings, integral domains, ideals, quotient rings, ring homomorphisms, polynomial rings, polynomial factorization, extension fields and properties of finite fields. Recommended background: MA 2073. Undergraduate credit may not be earned both for this course and for MA 3821. This course will be offered in 2015-16, and in alternating years thereafter.


Cat. I Principles of Real Analysis is a two-part course giving a rigorous presentation of the important concepts of classical real analysis. Topics covered in the sequence include: basic set theory, elementary topology of Euclidean spaces, metric spaces, compactness, limits and continuity, differentiation, Riemann-Stieltjes integration, infinite series, sequences of functions, and topics in multivariate calculus. Recommended background: at least one course focused on proof-based mathematics (e.g., MA 1971 Bridge to Higher Mathematics, MA1033 Theoretical Calculus III).


Cat. I MA 3832 is a continuation of MA 3831. For the contents of this course, see the description given for MA 3831. Recommended background: introductory knowledge in real analysis (e.g., MA 3831 Principles of Real Analysis I, or equivalent).





This course covers topics in loss models and risk theory as it is applied, under specified assumptions, to insurance. Topics covered include: economics of insurance, short term individual risk models, single period and extended period collective loss models, and applications. Recommended background: An introduction to probability (MA 2631 or equivalent).


Survival models are statistical models of times to occurrence of some event. They are widely used in areas such as the life sciences and actuarial science (where they model such events as time to death, or to the development or recurrence of a disease), and engineering (where they model the reliability or useful life of products or processes). This course introduces the nature and properties of survival models, and considers techniques for estimation and testing of such models using realistic data. Topics covered will be chosen from: parametric and nonparametric survival models, censoring and truncation, nonparametric estimation (including confidence intervals and hypothesis testing) using right-, left-, and otherwise censored or truncated data. Recommended background: An introduction to mathematical statistics (MA 3631 or equivalent).


This pass/fail graduation requirement will be offered every term, under the supervision of the actuarial professors. In order to receive a passing grade, students will need to complete some or all of the following: attend speaker talks, attend company visits to campus, take part and help out with Math Department activities, take part and help out with Actuarial Club activities, prepare for actuarial exams, or complete other activities as approved by the instructor(s). Recommended background: Interest in being an actuarial mathematics major.


Cat. II This course will introduce students to the top algorithms in applied mathematics. These algorithms have tremendous impact on the development and practice of modern science and engineering. Class discussions will focus on introducing students to the mathematical theory behind the algorithms as well as their applications. In particular, the course will address issues of computational efficiency, implementation, and error analysis. Algorithms to be considered may include the Krylov Subspace Methods, Fast Multipole Method, Monte Carlo Methods, Fast Fourier Transform, Kalman Filters and Singular Value Decomposition. Students will be expected to apply these algorithms to real-world problems e.g., image processing and audio compression (Fast Fourier Transform), recommendation systems (Singular Value Decomposition), electromagnetics or fluid dynamics (Fast Multipole Method, Krylov Subspace Methods, and Fast Fourier Transform), and the tracking and prediction of an object's position (Kalman Filters). In addition to studying these algorithms, students will learn about high performance computing and will have access to a machine with parallel and GPU capabilities to run code for applications with large data sets. Recommended background: Familiarity with matrix algebra and systems of equations (MA 2071, MA 2072, or equivalent), numerical methods for the solution of linear systems or differential equations (MA 3257, MA 3457, or equivalent), and concepts from probability (MA 2621, MA 2631, or equivalent). The ability to write computer programs in a scientific language is assumed.


Cat. II This course explores theoretical conditions for the existence of solutions and effective computational procedures to find these solutions for optimization problems involving nonlinear functions. Topics covered include: classical optimization techniques, Lagrange multipliers and Kuhn-Tucker theory, duality in nonlinear programming, and algorithms for constrained and unconstrained problems. Recommended background: Vector calculus at the level of MA 2251. This course will be offered in 2015-16, and in alternating years thereafter.


Cat. II This course develops probabilistic methods useful to planners and decision makers in such areas as strategic planning, service facilities design, and failure of complex systems. Topics covered include: decisions theory, inventory theory, queuing theory, reliability theory, and simulation. Recommended background: Probability theory at the level of MA 2621 or MA 2631. This course will be offered in 2015-16, and in alternating years thereafter.


Cat. I This course provides an introduction to the ideas and techniques of complex analysis that are frequently used by scientists and engineers. The presentation will follow a middle ground between rigor and intuition. Topics covered include: complex numbers, analytic functions, Taylor and Laurent expansions, Cauchy integral theorem, residue theory, and conformal mappings. Recommended background: MA 1024 and MA 2051.


Cat. II This course is concerned with the development and analysis of numerical methods for differential equations. Topics covered include: well-posedness of initial value problems, analysis of Euler's method, local and global truncation error, Runge-Kutta methods, higher order equations and systems of equations, convergence and stability analysis of one-step methods, multistep methods, methods for stiff differential equations and absolute stability, introduction to methods for partial differential equations. Recommended background: MA 2071 and MA 3457/CS 4033. An ability to write computer programs in a scientific language is assumed. This course will be offered in 2016-17, and in alternating years thereafter.


Cat. I Science and engineering majors often encounter partial differential equations in the study of heat flow, vibrations, electric circuits and similar areas. Solution techniques for these types of problems will be emphasized in this course. Topics covered include: derivation of partial differential equations as models of prototype problems in the areas mentioned above, Fourier Series, solution of linear partial differential equations by separation of variables, Fourier integrals and a study of Bessel functions. Recommended background: MA 1024 or and MA 2051.


Cat. II The first part of the course will cover the following topics: classification of partial differential equations, solving single first order equations by the method of characteristics, solutions of Laplace's and Poisson's equations including the construction of Green's function, solutions of the heat equation including the construction of the fundamental solution, maximum principles for elliptic and parabolic equations. For the second part of the course, the instructor may choose to expand on any one of the above topics. Recommended background: MA 2251 and MA 3832. This course will be offered in 2016-17, and in alternating years thereafter.


Cat. II This course provides students with knowledge and understanding of the applications of statistics in modern genetics and bioinformatics. The course generally covers population genetics, genetic epidemiology, and statistical models in bioinformatics. Specific topics include meiosis modeling, stochastic models for recombination, linkage and association studies (parametric vs. nonparametric models, family-based vs. population-based models) for mapping genes of qualitative and quantitative traits, gene expression data analysis, DNA and protein sequence analysis, and molecular evolution. Statistical approaches include log-likelihood ratio tests, score tests, generalized linear models, EM algorithm, Markov chain Monte Carlo, hidden Markov model, and classification and regression trees. Recommended background: MA 2612, MA 2631 (or MA 2621), and one or more biology courses. This course will be offered in 2015-16, and in alternating years thereafter.


Cat. I (14 week course) Intended for advanced undergraduates and beginning graduate students in the mathematical sciences, and for others intending to pursue the mathematical study of probability and statistics., this course begins by covering the material of MA 3613 at a more advanced level. Additional topics covered are: one-to-one and many-to-one transformations of random variablessampling distributions order statistics, limit theorems. Recommended background: MA 2631 or MA 3613, MA 3831, MA 3832.


Cat. I (14 week course) This course is designed to provide background in principles of statistics. Topics covered include: point and interval estimation sufficiency, completeness, efficiency, consistency the Rao-Blackwell Theorem and the Cramer-Rao bound minimum variance unbiased estimators, maximum likelihood estimators and Bayes estimators tests of hypothesis including uniformly most powerful, likelihood ratio, minimax and bayesian tests. Recommended background: MA 3631 or MA 4631.


Cat. I The focus of this class will be on statistical learning and the intersection of applied statistics and modeling techniques used to analyze and to make predictions and inferences from complex real-world data. Topics covered include: regression classification/clustering sampling methods (bootstrap and cross validation) and decision tree learning. Students may not receive credit for both MA463X and MA4635. Recommended background: Linear Algebra (MA2071 or equivalent), Applied Statistics and Regression (MA2612 or equivalent), Probability (MA2631 or equivalent). The ability to write computer programs in a scientific language is assumed.



Topics covered in this course would vary from one offering to the next. The purpose of this course will be to introduce actuarial topics that typically arise in the professional actuarial organization?s curriculum beyond the point where aspiring actuaries are still in college. Topics might include ratemaking, estimation of unpaid claims, equity linked insurance products, simulation, or stochastic modeling of insurance products. Recommended background: Could vary by the specific topics being covered, but would typically include an introduction to the theory of interest and an introduction to actuarial mathematics (MA 2211 and MA 3212 or equivalent)


A recurrence relation is an equation that expresses each element of a sequence as a function of the preceding ones. More precisely, in the case where only the immediately preceding element is involved, a recurrence relation has the form

It is easy to modify the definition for getting sequences starting from the term of index 1 or higher.

This defines recurrence relation of first order. A recurrence relation of order k has the form

Factorial Edit

The factorial is defined by the recurrence relation

and the initial condition

Logistic map Edit

An example of a recurrence relation is the logistic map:

with a given constant r given the initial term x0 each subsequent term is determined by this relation.

Solving a recurrence relation means obtaining a closed-form solution: a non-recursive function of n.

Fibonacci numbers Edit

The recurrence of order two satisfied by the Fibonacci numbers is the archetype of a homogeneous linear recurrence relation with constant coefficients (see below). The Fibonacci sequence is defined using the recurrence

Explicitly, the recurrence yields the equations

We obtain the sequence of Fibonacci numbers, which begins

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .

The recurrence can be solved by methods described below yielding Binet's formula, which involves powers of the two roots of the characteristic polynomial t 2 = t + 1 the generating function of the sequence is the rational function

Binomial coefficients Edit

with the base cases ( n 0 ) = ( n n ) = 1 <0>>=< binom >=1> . Using this formula to compute the values of all binomial coefficients generates an infinite array called Pascal's triangle. The same values can also be computed directly by a different formula that is not a recurrence, but that requires multiplication and not just addition to compute: ( n k ) = n ! k ! ( n − k ) ! . >=>.>

which can be simplified to

More generally: the k-th difference of the sequence an written as Δ k ( a n ) (a_)> is defined recursively as

(The sequence and its differences are related by a binomial transform.) The more restrictive definition of difference equation is an equation composed of an and its k th differences. (A widely used broader definition treats "difference equation" as synonymous with "recurrence relation". See for example rational difference equation and matrix difference equation.)

Actually, it is easily seen that,

Thus, a difference equation can be defined as an equation that involves an, an−1, an−2 etc. (or equivalently an, an+1, an+2 etc.)

Since difference equations are a very common form of recurrence, some authors use the two terms interchangeably. For example, the difference equation

is equivalent to the recurrence relation

Thus one can solve many recurrence relations by rephrasing them as difference equations, and then solving the difference equation, analogously to how one solves ordinary differential equations. However, the Ackermann numbers are an example of a recurrence relation that do not map to a difference equation, much less points on the solution to a differential equation.

See time scale calculus for a unification of the theory of difference equations with that of differential equations.

Summation equations relate to difference equations as integral equations relate to differential equations.

From sequences to grids Edit

Single-variable or one-dimensional recurrence relations are about sequences (i.e. functions defined on one-dimensional grids). Multi-variable or n-dimensional recurrence relations are about n-dimensional grids. Functions defined on n-grids can also be studied with partial difference equations. [2]

Solving homogeneous linear recurrence relations with constant coefficients Edit

Roots of the characteristic polynomial Edit

An order-d homogeneous linear recurrence with constant coefficients is an equation of the form

where the d coefficients ci (for all i) are constants, and c d ≠ 0 eq 0> .

A constant-recursive sequence is a sequence satisfying a recurrence of this form. There are d degrees of freedom for solutions to this recurrence, i.e., the initial values a 0 , … , a d − 1 ,dots ,a_> can be taken to be any values but then the recurrence determines the sequence uniquely.

The same coefficients yield the characteristic polynomial (also "auxiliary polynomial")

whose d roots play a crucial role in finding and understanding the sequences satisfying the recurrence. If the roots r1, r2, . are all distinct, then each solution to the recurrence takes the form

where the coefficients ki are determined in order to fit the initial conditions of the recurrence. When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as (xr) 3 , with the same root r occurring three times, then the solution would take the form

As well as the Fibonacci numbers, other constant-recursive sequences include the Lucas numbers and Lucas sequences, the Jacobsthal numbers, the Pell numbers and more generally the solutions to Pell's equation.

For order 1, the recurrence

has the solution an = r n with a0 = 1 and the most general solution is an = kr n with a0 = k. The characteristic polynomial equated to zero (the characteristic equation) is simply tr = 0.

Solutions to such recurrence relations of higher order are found by systematic means, often using the fact that an = r n is a solution for the recurrence exactly when t = r is a root of the characteristic polynomial. This can be approached directly or using generating functions (formal power series) or matrices.

Consider, for example, a recurrence relation of the form

When does it have a solution of the same general form as an = r n ? Substituting this guess (ansatz) in the recurrence relation, we find that

must be true for all n > 1.

Dividing through by r n−2 , we get that all these equations reduce to the same thing:

which is the characteristic equation of the recurrence relation. Solve for r to obtain the two roots λ1, λ2: these roots are known as the characteristic roots or eigenvalues of the characteristic equation. Different solutions are obtained depending on the nature of the roots: If these roots are distinct, we have the general solution

while if they are identical (when A 2 + 4B = 0 ), we have

This is the most general solution the two constants C and D can be chosen based on two given initial conditions a0 and a1 to produce a specific solution.

In the case of complex eigenvalues (which also gives rise to complex values for the solution parameters C and D), the use of complex numbers can be eliminated by rewriting the solution in trigonometric form. In this case we can write the eigenvalues as λ 1 , λ 2 = α ± β i . ,lambda _<2>=alpha pm eta i.> Then it can be shown that

can be rewritten as [4] : 576–585

Here E and F (or equivalently, G and δ) are real constants which depend on the initial conditions. Using

one may simplify the solution given above as

where a1 and a2 are the initial conditions and

In this way there is no need to solve for λ1 and λ2.

In all cases—real distinct eigenvalues, real duplicated eigenvalues, and complex conjugate eigenvalues—the equation is stable (that is, the variable a converges to a fixed value [specifically, zero]) if and only if both eigenvalues are smaller than one in absolute value. In this second-order case, this condition on the eigenvalues can be shown [5] to be equivalent to |A| < 1 − B < 2, which is equivalent to |B| < 1 and |A| < 1 − B.

The equation in the above example was homogeneous, in that there was no constant term. If one starts with the non-homogeneous recurrence

with constant term K, this can be converted into homogeneous form as follows: The steady state is found by setting bn = bn−1 = bn−2 = b* to obtain

Then the non-homogeneous recurrence can be rewritten in homogeneous form as

which can be solved as above.

The stability condition stated above in terms of eigenvalues for the second-order case remains valid for the general n th -order case: the equation is stable if and only if all eigenvalues of the characteristic equation are less than one in absolute value.

Given a homogeneous linear recurrence relation with constant coefficients of order d, let p(t) be the characteristic polynomial (also "auxiliary polynomial")

such that each ci corresponds to each ci in the original recurrence relation (see the general form above). Suppose λ is a root of p(t) having multiplicity r. This is to say that (t−λ) r divides p(t). The following two properties hold:

As a result of this theorem a homogeneous linear recurrence relation with constant coefficients can be solved in the following manner:

  1. Find the characteristic polynomial p(t).
  2. Find the roots of p(t) counting multiplicity.
  3. Write an as a linear combination of all the roots (counting multiplicity as shown in the theorem above) with unknown coefficients bi. a n = ( b 1 λ 1 n + b 2 n λ 1 n + b 3 n 2 λ 1 n + ⋯ + b r n r − 1 λ 1 n ) + ⋯ + ( b d − q + 1 λ ∗ n + ⋯ + b d n q − 1 λ ∗ n ) =left(b_<1>lambda _<1>^+b_<2>nlambda _<1>^+b_<3>n^<2>lambda _<1>^+cdots +b_n^lambda _<1>^ ight)+cdots +left(b_lambda _<*>^+cdots +b_n^lambda _<*>^ ight)> This is the general solution to the original recurrence relation. (q is the multiplicity of λ*)
  4. Equate each a 0 , a 1 , … , a d ,a_<1>,dots ,a_> from part 3 (plugging in n = 0, . d into the general solution of the recurrence relation) with the known values a 0 , a 1 , … , a d ,a_<1>,dots ,a_> from the original recurrence relation. However, the values an from the original recurrence relation used do not usually have to be contiguous: excluding exceptional cases, just d of them are needed (i.e., for an original homogeneous linear recurrence relation of order 3 one could use the values a0, a1, a4). This process will produce a linear system of d equations with d unknowns. Solving these equations for the unknown coefficients b 1 , b 2 , … , b d ,b_<2>,dots ,b_> of the general solution and plugging these values back into the general solution will produce the particular solution to the original recurrence relation that fits the original recurrence relation's initial conditions (as well as all subsequent values a 0 , a 1 , a 2 , … ,a_<1>,a_<2>,dots > of the original recurrence relation).

The method for solving linear differential equations is similar to the method above—the "intelligent guess" (ansatz) for linear differential equations with constant coefficients is e λx where λ is a complex number that is determined by substituting the guess into the differential equation.

This is not a coincidence. Considering the Taylor series of the solution to a linear differential equation:

it can be seen that the coefficients of the series are given by the n th derivative of f(x) evaluated at the point a. The differential equation provides a linear difference equation relating these coefficients.

This equivalence can be used to quickly solve for the recurrence relationship for the coefficients in the power series solution of a linear differential equation.

The rule of thumb (for equations in which the polynomial multiplying the first term is non-zero at zero) is that:

Example: The recurrence relationship for the Taylor series coefficients of the equation:

n ( n − 1 ) f [ n + 1 ] + 3 n f [ n + 2 ] − 4 f [ n + 3 ] − 3 n f [ n + 1 ] − f [ n + 2 ] + 2 f [ n ] = 0

− 4 f [ n + 3 ] + 2 n f [ n + 2 ] + n ( n − 4 ) f [ n + 1 ] + 2 f [ n ] = 0.

This example shows how problems generally solved using the power series solution method taught in normal differential equation classes can be solved in a much easier way.

Example: The differential equation

The conversion of the differential equation to a difference equation of the Taylor coefficients is

a f [ n + 2 ] + b f [ n + 1 ] + c f [ n ] = 0.

It is easy to see that the nth derivative of e ax evaluated at 0 is a n .

Solving via linear algebra Edit

A linearly recursive sequence y of order n

Expanded with n−1 identities of kind y n − k = y n − k =y_> , this n-th order equation is translated into a matrix difference equation system of n first-order linear equations,

This description is really no different from general method above, however it is more succinct. It also works nicely for situations like

where there are several linked recurrences. [6]

Solving with z-transforms Edit

Certain difference equations - in particular, linear constant coefficient difference equations - can be solved using z-transforms. The z-transforms are a class of integral transforms that lead to more convenient algebraic manipulations and more straightforward solutions. There are cases in which obtaining a direct solution would be all but impossible, yet solving the problem via a thoughtfully chosen integral transform is straightforward.

Solving non-homogeneous linear recurrence relations with constant coefficients Edit

If the recurrence is non-homogeneous, a particular solution can be found by the method of undetermined coefficients and the solution is the sum of the solution of the homogeneous and the particular solutions. Another method to solve a non-homogeneous recurrence is the method of symbolic differentiation. For example, consider the following recurrence:

This is a non-homogeneous recurrence. If we substitute nn+1, we obtain the recurrence

Subtracting the original recurrence from this equation yields

This is a homogeneous recurrence, which can be solved by the methods explained above. In general, if a linear recurrence has the form

is the generating function of the inhomogeneity, the generating function

of the non-homogeneous recurrence

with constant coefficients ci is derived from

If P(x) is a rational generating function, A(x) is also one. The case discussed above, where pn = K is a constant, emerges as one example of this formula, with P(x) = K/(1−x). Another example, the recurrence a n = 10 a n − 1 + n =10a_+n> with linear inhomogeneity, arises in the definition of the schizophrenic numbers. The solution of homogeneous recurrences is incorporated as p = P = 0.

Solving first-order non-homogeneous recurrence relations with variable coefficients Edit

Moreover, for the general first-order non-homogeneous linear recurrence relation with variable coefficients:

there is also a nice method to solve it: [7]

If we apply the formula to a n + 1 = ( 1 + h f n h ) a n + h g n h =(1+hf_)a_+hg_> and take the limit h→0, we get the formula for first order linear differential equations with variable coefficients the sum becomes an integral, and the product becomes the exponential function of an integral.

Solving general homogeneous linear recurrence relations Edit

Many homogeneous linear recurrence relations may be solved by means of the generalized hypergeometric series. Special cases of these lead to recurrence relations for the orthogonal polynomials, and many special functions. For example, the solution to

the confluent hypergeometric series. Sequences which are the solutions of linear difference equations with polynomial coefficients are called P-recursive. For these specific recurrence equations algorithms are known which find polynomial, rational or hypergeometric solutions.

Solving first-order rational difference equations Edit

Stability of linear higher-order recurrences Edit

The linear recurrence of order d,

The recurrence is stable, meaning that the iterates converge asymptotically to a fixed value, if and only if the eigenvalues (i.e., the roots of the characteristic equation), whether real or complex, are all less than unity in absolute value.

Stability of linear first-order matrix recurrences Edit

In the first-order matrix difference equation

with state vector x and transition matrix A, x converges asymptotically to the steady state vector x* if and only if all eigenvalues of the transition matrix A (whether real or complex) have an absolute value which is less than 1.

Stability of nonlinear first-order recurrences Edit

Consider the nonlinear first-order recurrence

This recurrence is locally stable, meaning that it converges to a fixed point x* from points sufficiently close to x*, if the slope of f in the neighborhood of x* is smaller than unity in absolute value: that is,

A nonlinear recurrence could have multiple fixed points, in which case some fixed points may be locally stable and others locally unstable for continuous f two adjacent fixed points cannot both be locally stable.

A nonlinear recurrence relation could also have a cycle of period k for k > 1. Such a cycle is stable, meaning that it attracts a set of initial conditions of positive measure, if the composite function

with f appearing k times is locally stable according to the same criterion:

where x* is any point on the cycle.

In a chaotic recurrence relation, the variable x stays in a bounded region but never converges to a fixed point or an attracting cycle any fixed points or cycles of the equation are unstable. See also logistic map, dyadic transformation, and tent map.

When solving an ordinary differential equation numerically, one typically encounters a recurrence relation. For example, when solving the initial value problem

with Euler's method and a step size h, one calculates the values

Systems of linear first order differential equations can be discretized exactly analytically using the methods shown in the discretization article.

Biology Edit

Some of the best-known difference equations have their origins in the attempt to model population dynamics. For example, the Fibonacci numbers were once used as a model for the growth of a rabbit population.

The logistic map is used either directly to model population growth, or as a starting point for more detailed models of population dynamics. In this context, coupled difference equations are often used to model the interaction of two or more populations. For example, the Nicholson–Bailey model for a host-parasite interaction is given by

with Nt representing the hosts, and Pt the parasites, at time t.

Integrodifference equations are a form of recurrence relation important to spatial ecology. These and other difference equations are particularly suited to modeling univoltine populations.

Computer science Edit

Recurrence relations are also of fundamental importance in analysis of algorithms. [8] [9] If an algorithm is designed so that it will break a problem into smaller subproblems (divide and conquer), its running time is described by a recurrence relation.

A simple example is the time an algorithm takes to find an element in an ordered vector with n elements, in the worst case.

A naive algorithm will search from left to right, one element at a time. The worst possible scenario is when the required element is the last, so the number of comparisons is n .

A better algorithm is called binary search. However, it requires a sorted vector. It will first check if the element is at the middle of the vector. If not, then it will check if the middle element is greater or lesser than the sought element. At this point, half of the vector can be discarded, and the algorithm can be run again on the other half. The number of comparisons will be given by

Digital signal processing Edit

In digital signal processing, recurrence relations can model feedback in a system, where outputs at one time become inputs for future time. They thus arise in infinite impulse response (IIR) digital filters.

For example, the equation for a "feedforward" IIR comb filter of delay T is:

Economics Edit

Recurrence relations, especially linear recurrence relations, are used extensively in both theoretical and empirical economics. [10] [11] In particular, in macroeconomics one might develop a model of various broad sectors of the economy (the financial sector, the goods sector, the labor market, etc.) in which some agents' actions depend on lagged variables. The model would then be solved for current values of key variables (interest rate, real GDP, etc.) in terms of past and current values of other variables.


In light of the Covid-19 pandemic the University has revised its courses to incorporate the ‘Hybrid Learning Experience’ in a departure from previous academic years and previously published information. The University has changed the delivery (and in some cases the content) of its programmes. Further information on the general principles of hybrid learning can be found at: Hybrid learning experience | University of Surrey.

We have updated key module information regarding the pattern of assessment and overall student workload to inform student module choices. We are currently working on bringing remaining published information up to date to reflect current practice in time for the start of the academic year 2021/22.

This means that some information within the programme and module catalogue will be subject to change. Current students are invited to contact their Programme Leader or Academic Hive with any questions relating to the information available.

This module builds on the differential equation aspects of the level 1 modules Calculus and Linear Algebra and considers qualitative and quantitative aspects of Ordinary Differential Equations.

9.1E: Introduction to Linear Higher Order Equations (Exercises) - Mathematics

Differential Equations (Solution) William Trench [PDF]

Elementary Differential Equations with Boundary Value Problems (Solution Manual) by William F. Trench

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About this book :-
Elementary Differential Equations with Boundary Value Problems (Solution Manual) written by William F. Trench .
Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation.
An elementary text should be written so the student can read it with comprehension without too much pain. The author has tried to put myself in the student’s place, and have chosen to err on the side of too much detail rather than not enough.
An elementary text can’t be better than its exercises. This text includes 2041 numbered exercises, many with several parts. They range in difficulty from routine to very challenging.
An elementary text should be written in an informal but mathematically accurate way, illustrated by appropriate graphics. The author has tried to formulate mathematical concepts succinctly in language that students can understand. The author has minimized the number of explicitly stated theorems and definitions, preferring to deal with concepts in a more conversational way, copiously illustrated by 299 completely worked out examples. Where appropriate, concepts and results are depicted in 188 figures.
(William F. Trench)

Book Detail :-
Title: Elementary Differential Equations with Boundary Value Problems (Solution Manual)
Author(s): William F. Trench
Publisher: Brooks/Cole Thomson Learning
Year: 2013
Pages: 288
Type: PDF
Language: English
Country: US
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About Author :- The author William F Trench , joined the Trinity faculty as the Cowles Distinguished Professor in 1986. He had previously served as professor of mathematics at Drexel University from 1964 to 1986. Prior to entering academia, Trench was employed as an applied mathematician by RCA, Philco, and General Electric.
His main research interests were in linear algebra and ordinary differential equations. He was the recipient of multiple National Science Foundation (NSF) grants. In 1989 he received a $35,575 grant in support of a project on "Numerical Solution of Spectral Problems for Efficiently Structured Hermitian Matrices. Trench was the author of three textbooks and more than 120 research papers. All three of his textbooks are available via open access on the Trinity Digital Commons, where they have been downloaded more than 81,000 times. His works are the most downloaded files across the entire Trinity faculty. Although he retired from teaching in 1997, he continued to publish research with articles appearing in leading journals as recently as 2014.
His teaching experience spanned nearly 35 years and included a broad spectrum of mathematics courses to undergraduates and graduate students. At Trinity students found him to be interesting and intriguing and especially appreciated his sense of humor.
He earned B.Sc bachelor's in mathematics from Lehigh University, M.Sc and Ph.D. in mathematics from the University of Pennsylvania. He remaind also a member of the Society for Industrial and Applied Mathematics, the American Mathematical Society, and Phi Beta Kappa.

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Book Contents :-
Elementary Differential Equations with Boundary Value Problems (Solution Manual) written by William F. Trench cover the following topics. '
1. Introduction
2. First Order Equations
3. Numerical Methods
4. Applications of First Order Equations1em
5. Linear Second Order Equations
6. Applcations of Linear Second Order Equations
7. Series Solutions of Linear Second Order Equations
8. Laplace Transforms
9. Linear Higher Order Equations
10. Linear Systems of Differential Equations
11. Boundary Value Problems and Fourier Expansions
12. Fourier Solutions of Partial Differential Equations
13. Boundary Value Problems for Second Order Linear Equations

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1 . An Introduction to Mathematica

One of the most important aspects of any software package is how readily one is able to access the information from the output of a command and redirect it as input to another. For example, throughout the text we will make extensive use of the differential equation solver of Mathematica in various applications, where monitoring the evolution of certain variables is key to our understanding of a physical model. Typically, we would like to graph some or all of the variables we obtain from solving a system of differential equations as time varies, or graph one variable versus another, or integrate and differentiate the output in order to construct physical quantities that have natural interpretations. This chapter is an introduction to how one can accomplish such tasks with simple examples that are similar to the actual circumstances we encounter in the remainder of the text.

The material presented here is intended primarily to be used as reference for the exercises in the upcoming chapters. As a result, during the first reading some of the mathematical language may be unfamiliar. It is hoped that this chapter will become more useful to the reader as one proceeds with the mathematical concepts of the future chapters but returns frequently here to review the relevant syntax.

Using Mathematica one is able to

  1. Integrate and differentiate symbolically rather complicated expressions
  2. Generate graphics in two and three dimensions
  3. Simplify trigonometric and algebraic expressions
  4. Solve linear and nonlinear differential equations
  5. Determine the Laplace and Fourier transforms of functions

1.2 A Session in Mathematica

In the Notebook version of Mathematica one either selects the appropriate icon and executes the application (this is the case on PCs and Apple computers) or enters at the system prompt, as is the case on SUN workstations. After some initialization, a window opens in which one is allowed to enter commands. Commands are entered at the keyboard and executed by pressing the Shift and Return (or Enter ) keys simultaneously . After the first command is executed In[1]:= will appear on the screen.

As mentioned earlier, commands are simply entered at the keyboard. For example, to find the roots of the polynomial

f(x) = x 2 -4x+3,
type The program will respond with that is, 1 and 3 are the roots of the polynomial. : Mathematica distinguishes between x = y and x = = y. In the first expression, x = y, x is assigned the value y, while in the second expression Mathematica checks to see if x and y can be equal to each other, that is, it verifies that x and y are compatible objects, and other than that it does not take any action. We will use to define equations. : Multiplication in Mathematica can be entered either by leaving a blank space between the operands (for example, a x represents ax) or by putting an asterisk between the terms (that is, a*x ). If a is a number, however, Mathematica interprets the combination of a number adjacent to a variable as multiplication. For instance, 2x is understood as 2 times x. This convention does not apply to symbols: the expression ax is interpreted as the variable whose name is ax and not as the product of the variables a and x. : We will ignore the prompts In[]:= and Out[]:= for the rest of this discussion. The first important feature of Mathematica is that this program is case sensitive that is, it distinguishes between solve and Solve in the first command that we entered. All functions and programs that are internally known to Mathematica must be capitalized. Thus, Solve is a subroutine in Mathematica that is capable of finding roots of polynomials, while solve does not have a special meaning unless the user has defined it previously. Similarly, Mathematica understands that Sin is the usual sine function while sin is a name of a variable that is unknown in this session.

Functions in Mathematica are delimited by [ ] and not by ( ). So Sin[x] is the usual sine function while Sin(x) is the variable Sin multiplied by the variable x . The operator Solve is an example of a function with two arguments the first is used to define the equation whose solutions we are interested in, and the second is the variable with respect to which the roots should be computed. All of these arguments are grouped by [ ]. To get a better understanding for the usage of arguments in Mathematica let us try the following two examples: and Do the outputs make sense? In b the symbol x is the variable with respect to which the roots of the polynomial are determined and a is just a parameter, while in c the roles of x and a are reversed. The above examples show the most important attribute of Mathematica : its ability to do symbolic manipulation. In b Mathematica does not need to know the value for a in order to find the roots of the second-order polynomial. This is just one example among many in which Mathematica uses its logical power and is able to find the answer to certain questions. There are limitations to this ability, unfortunately. Try finding the roots of the following polynomial on Mathematica :

x 5 - a x 4 + 3 x 3 + 2 x 2 + x -1 = 0.
Mathematica 's response is its way of saying that it does not know how to find symbolically the roots of fifth-order polynomials. This answer is in agreement with a celebrated theorem in group theory that confirms our inability to write formulas for roots of general polynomials with degrees larger than or equal to five. On the other hand, by simply drawing the graph of this fifth-order polynomial for a fixed a, we recognize that it has some real roots. Mathematica is able to find these roots numerically , that is, by approximate methods. To accomplish this one needs to furnish a concrete value of the parameter a in the definition of the polynomial. Try Mathematica finds all five roots of this polynomial: NSolve is a program based on a numerical approximation technique that is capable of finding solutions to certain equations in particular, it is capable of finding roots of polynomials.

  1. Use ? (see the next section for more detail) with Solve and NSolve , and familiarize yourself with the syntax of these commands.
  2. Use Solve or NSolve to determine the roots or zeros of the following expressions.
    1. a x 2 + bx +c
    2. a x 3 + b x 2 + c x + d
    3. x 2 + 1
    4. x 3 + 1
    5. x 1/3 - x + 1
    6. sinx - 1/3
    7. sin 2 x - 1/3
    8. sinx 2 - 1/3
    9. sinx - x. Familiarize yourself with the syntax of the FindRoot command by using ? . Then use FindRoot in this problem.
    1. tan x - 3x + 1
    2. The system of algebraic equations
      1. 3 x - 2 y = 2, x + y = 7
      2. a x - y = 0, x + a y = 1
      3. x 2 - y 2 = 1, x 2 + y 2 = 4
      4. x 3 - y 3 = 1, x 2 - 3 x y + y 2 = 8
      5. a x + y +z = 1, x - y + 2 z = 0, 2x + 3 y - z = 2
      6. 3 x 2 - 4y 2 + 3 z = 1, x + y + z = 0, z 3 - x 2 y = -1

      1.3 The Help Command in Mathematica

      1.4 Factoring and Simplification

      Some of the elementary operations that Mathematica is capable of performing include expanding, factoring, and simplifying expressions. For instance, the command for expanding
      (a+b+c) 3
      is The first command clears any previous values assigned to a , b , and c . Mathematica responds with We can now factor the above expression (assuming that it is stored in Out[4] of the present session) by entering thereby recovering the original expression. We note that, in place of Factor[Out[4]] %, we could use to reach the same result if the expression on which Factor acts is the latest output of the session.

      The operations of Expand , Factor , and Simplify are very useful when we wish to prove the type of identities commonly encountered in elementary algebra. For example, recall the identity

      (a+b) 2 - (a-b) 2 = 4 a b
      for any two parameters a and b. To verify this identity on Mathematica we enter If we repeat the above command but replace Expand with Factor , we obtain the same result. Now, let us try the previous three commands on
      a 2 - 2 a b + b 2 .
      Factor and Simplify give us the appropriate alternative expression for a 2 - 2 ab + b 2 , while Expand leaves the expression unchanged. Next, we apply these three commands to the expression
      1/(x+t )- 1/(x-t)
      to get a better sense for the range of capabilities of each the above internal functions: Expand returns the original expression whereas Factor returns while Simplify gives us Mathematica is also capable of manipulating trigonometric functions. For example, returns the result we expect from elementary trigonometry. To get a sense for the power of Mathematica 's Simplify command let us try Does the answer make sense?

      1. Verify the following identities in Mathematica .
        1. sin 2x = 2sin x cos x
        2. cos 2x = 2 cos 2 x - 1
        3. sin 3x = 4 sin 3 x - 3 sin x
        4. tan(x + y) = (tan x + tan y)/(1 - tan x tan y)
        1. cot(x + y), cot x, and cot y
        2. sin(4x), sin x, and cos x
        3. cos x and cos x/2
        4. tan 2x and cos x
        1. cos(x+y)+cos(x-y)
        2. cos(x+y)-cos(x-y)
        3. cos(x+y)+sin(x+y)
        4. cos 2 2x - sin 2 2x
        5. cos 2 a x - sin 2 a x
        6. cos 3 x - sin 3 x (Try Factor followed by Simplify .)

        1.5 Function Definition

        It is also possible to define functions of more than one variable in Mathematica . For example, the second-order polynomial g(x) = x 2 - a x + 3 is defined by and the operation of finding its roots is accomplished by entering Many internal functions in Mathematica take as input expressions that involve functions. We have already seen the example of f in (1.1) and the internal function Solve . Another example arises with FindRoot , a variant of the Solve command. This internal function computes roots of functions numerically and is preferable to Solve in cases where the function f is rather complicated and one must resort to approximate methods to seek its roots. Its syntax requires specifying an initial guess for a root. For instance, determines a root of f by applying a root-finding algorithm to f, based on Newton's method, starting the algorithm at the point x = 3. Mathematica returns a good (although not a very good) approximation to the exact value p . There are, however, two options available to FindRoot that render the approximate value much closer to the exact value (try ?? FindRoot for the list of options). They are MaxIterations and WorkingPrecision . When we try we end up with which is accurate to eight digits. Some of the trailing digits in the above answer may differ on your machine. The above calculation was performed on a SUN workstation and PC 486 using version 3.0 of Mathematica .

        The reader may recall that Newton's method is based on the tangent vector approximation of f, which requires differentiating f. In cases where it is cumbersome to compute the derivative of f, the secant method, which replaces the tangent vector with a chord passing through two points on the graph of f, is preferable. One of the options in FindRoot allows for specifying two starting points, at which time a variant of the secant method is called upon. For example, uses function evaluations at x = 3 and x = 4 to start the approximation algorithm.

        1. Define the following functions in Mathematica and evaluate them at the specified points.
          1. f(x) = (1-x)/(1+x): x = 0, 0.5, and p
          2. f(x) = log(x + [ Ö (1 - x 2 )]): x = 0, 0.1, 0.2, and 0.3
          1. f(x) = x - 2/x
          2. f(x) = x 2 - 2/x
          3. f(x) = (x 2 -1)(x-2) + x
          4. x sin x + cos x

          1.6 Differentiation and Integration

          is able to differentiate and integrate functions symbolically. The derivative of a function such as x sin 2 x is found by while its integral is determined by entering The D and Integrate commands of Mathematica use the basic properties of the differentiation and integration operators, such as the linearity property, to reduce complicated computations to a series of simpler ones. These properties are then combined with elaborate tables of known derivatives and integrals that allow this software to reach its goal successfully. The power of this software is particularly noticed in the case of integration, where we recall from elementary calculus that we often have to resort to methods such as partial fractions, or special substitutions, or integration by parts to reduce the integrand to a manageable expression. For example, to evaluate the integral
          1/(1+x 4 )dx
          using standard tables, we must first note that 1+x 4 factors into
          (1- Ö 2x+x 2 )(1+ Ö 2x + x 2 )
          and apply the method of partial fraction before using the table of integration. On the other hand, yields The definite integral _0^1 dx is determined in a similar fashion: which results in To get a decimal approximation to the above value, we apply the N operation to it: recalling that % stands for the previous output. The new output is Mathematica is also capable of performing numerical integration. The internal function NIntegrate returns an approximate value of the function on which it operates. For example, to evaluate () numerically we enter in Mathematica , which compares well with the result of the numerical value we obtained after evaluating this integral exactly.

          In spite our best effort, it is fair to say the class of functions whose anti-derivative we are able to write down explicitly is rather small. If we just started to list functions at random, we could quickly generate functions whose anti-derivatives are either cumbersome to evaluate or actually impossible to express in terms of elementary functions of calculus. Many functions in mathematical physics fall in the latter category, among them e -x2 , sin(x 2 ), and frac1 Ö <1-m sin 2 x>. As a result of efforts of many mathematical analysts in the past few hundred years, properties of such anti-derivatives are tabulated, which are now generally available in most computer algebras, including in Mathematica . For example, the function

          f(x) = ó


          e -t2 dt
          can be accessed as Mathematica responds with The internal function Erf[x] is called the error function . It appears prominently in probability theory, among other branches of mathematics. We can now manipulate f just like any other function defined in Mathematica . Another example is
          g(x) = ó


          sint 2 dt.
          When we enter we get where the function FresnelS is called the Fresnel Integral, which has applications in the theory of light diffraction. A third example is h(x) = _0^1 dt. This time when we try we get the answer being given in terms of the Elliptic Function of the First Kind . We can obtain numerical values and graphical data from this function as before. For example, let us try Mathematica responds with where a numerical approximation to the integral in h is obtained. : The function h defined in () appears naturally in the context of the period of vibration of a nonlinear pendulum. Let us proceed a bit further with the last example. Since h defines a function in Mathematica , we should be able to differentiate it with respect to x. Let i(x) = h'(x), which we evaluate by The output is To evaluate i at x = 0.1: which results in is capable of determining partial derivatives as well as multiple integrals. Because we will address these topics in detail later in the text, we postpone their treatment in Mathematica at this time.

          1. Differentiate the following functions.
            1. f(x) = log x/(x+1)
            2. f(x) = sin 3 4x cos 5 7(x 2 -2x +1)
            3. f(x) = x x-1
            1. f(x) = x/(x-1)
            2. f(x) = x sin x
            3. f(x) = x 2 sin x
            4. f(x) = x 10 sin x
            5. f(x) = e x sin x (e x is Exp[x] in Mathematica )
            6. f(x) = sin 2 x
            7. f(x) = sin(x 2 )
            8. f(t) = t e t^2
            9. f(s) = e s^2
            1. ò - ¥ ¥ 1/(1 + x 2) dx (Ans: Integrate[1/(1+x^2), x, -Infinity, Infinity )
            2. ò 0 ¥ e -t^2 dt
            3. ò 0 ¥ e -a t^2 dt, a is a parameter
            4. ò 0 ¥ e -s^t sint dt (This is the Laplace transform of sin t.)

            a(x) f(x, t) dt ) = f(x,b(x))-f(x,a(x))+ ó
            õ b(x)

            1.7 Two-Dimensional Graphics

            In applications we often need to plot several graphs on the same screen. For instance, as we will see in the later chapters, the set of functions f(x) = sinn x represents a set of basic functions in terms of which we expand Fourier series of a large class of functions. Each sinn x, with n a positive integer, represents a mode of oscillation with period frac2 p n. By graphing these functions one sees how the different modes compare to one another. There are several ways to draw the graphs of sinn x, with n = 1, 2 and 3 on the same screen. One way is to draw the graphs of each mode separately, and then to combine them by using Show : Each one of the first three lines draws a separate graph. The last line combines the three graphs on a new screen.

            A second way of getting the same result is to use the following syntax: Figure shows the output. Figure Figure 1: The output for Plot[Evaluate[Table[Sin[n x], n, 1, 3]], x, 0, 2 Pi]. To obtain a hardcopy of any graph we produced in Mathematica , we need to take into account the specific features of the software and the platform on which it is installed. For example, to get a hardcopy of the above graph in a Notebook session, we must first highlight the cell containing the graph and then select Print from the File Menu . On the other hand, when the stand-alone version of Mathematica is accessed on a SUN workstation, the command PSPrint will do the job. For example, assuming that the above graph is stored in Out[24], try or, simply, if the graph is the latest output.

            Another way to draw the graphs of sinnx is by using GraphicsArray : Now, the above three graphs are plotted separately but on the same screen (cf. Figure ). This form of graphics is useful for displaying wave motion. Figure Figure 2: The output of the Show[GraphicsArray[. ]] command.

            1.7.1 Curves in the Plane

            Curves are the geometric manifestations of particle motions in a domain. Typically in this text curves represent motions of fluid particles in force fields and therefore, are parametrized by time. The position of a particle at time t is represented by a vector r (t) whose endpoint denotes the location of the particle at t. For example,
            r (t) = á 2sin t, 2cos t ñ
            defines a curve in the plane R 2 , where the x and y components of each point are 2sint and 2cost, respectively. Since x 2 + y 2 = 4, this curve is a circle of radius 2. To draw its graph, we apply ParametricPlot to r : One of the options of Show is AspectRatio . Hence, displays a circle with the 1:1 aspect ratio. We have the option of combining the above two commands into one as follows: ParametricPlot has several options, among them PlotPoints and PlotStyle . The PlotPoints option is particularly useful when the curve's domain is highly oscillatory because this option designates the number of points at which the parametrization r is to be evaluated. The default value of PlotPoints is 25. Compare the outputs of and We will come back to parametrization and ParametricPlot in more detail when we review curves, and the concepts of velocity and acceleration, in the context of vector calculus.
            1. Plot the graphs of the following functions.
              1. f(x) = sin(5x). What is the period of this function that is, what is the smallest value of T > 0 for which f(x+T) = f(x)?
              2. g(x) = sin(2x) + 3sin(3x). What is the period of this function?
              3. h(x) = sinx + sin Ö 2 x. Draw the graph of this function on the intervals (0,5), (0,10), and (0,50). Do these graphs give any indication as to whether h is periodic or not? Why?
              1. Plot the graph of this function. Use the option PlotRange - > All with Plot to get the entire range of the function.
              2. Compare this graph to the graphs of the functions g(x) = f(x+4) and h(x) = f(x/0.5) over the same domain. Is there a scale change among these three graphs?
              3. Draw on the same screen the graph of (f(x-2 t)+f(x+2 t)) for t ranging between 0 and 2 at increments of 0.25. Do this part by first defining () as a new function g(x,t) and then using to draw all of the graphs on the same screen. Experiment with the options GridLines and Frame until you obtain a figure similar to Figure . Is there a scale change in these graphs?
              1. r (t) = á sin 2 t, cos 2 t ñ t Î (0, 2 p )
              2. r (t) = á sin 5 t, cos 5 t ñ t Î (0, 2 p )
              3. r (t) = á sin 3 t, cos 2 t ñ t Î (0, 2 p )
              4. r (t) = á sin 3 t, cos2t ñ t Î (0, 2 p )
              5. r (t) = á sin 5 t, cos(1+2t) ñ t Î (0, 2 p )

              1.8 Three-Dimensional Graphics

              The syntax for plotting graphs of surfaces in three dimensions is very similar to two-dimensional graphics. For instance, produces the surface of the function f defined by f(x,y) = sinx cosy in the domain (0, p )×(0, p ). Similar to Plot , Plot3D has several options, which we examine in the context of the graph of a ``sombrero": The mathematical equation that describes this surface is
              f(x,y) = sinr/r, where r = _____
              Ö x 2 +y 2
              First, we define the polar radius r: and then plot the function f by We now take two steps to improve the picture on the screen. First, we use PlotRange to force Mathematica to show us the entire range of the plot: The second step is to use PlotPoints with Plot3D to require Mathematica to use more points on the x- and y- axes in its plotting routine: Figure Figure 4: The graph of the ``sombrero." The ParametricPlot3D is the analogue of ParametricPlotParametricPlot for curves whose range is in the three-dimensional space R 3 . For example, the graph of a helix whose parametrization is
              r (t) = á sin3t, cos3t, t ñ ,
              with t Î (0, 2 p ), is obtained from One can set the aspect ratio of the graph to any desired value using the AspectRatio option with either ParametricPlot3D or Show .

              We can also plot the graphs of surfaces with ParametricPlot3D . The parametrization of a surface, by definition, requires two independent parameters. For example, the surface whose equation is given by z = x 2 +y 2 within the domain (x, y) Î (-2, 2)×(-2, 2) can also be viewed as the set of points (x, y, x 2 + y 2 ). Here, the two independent parameters are x and y, each of which takes on values in the interval (-2,2). ParametricPlot3D 's syntax for displaying this surface is The surfaces we have considered thus far have the property that one could write down an explicit formula that relates one of the coordinates of the set of points on the surface to the remaining two. For example, the set of points (x, y, x 2 + y 2 ), with (x,y) Î (a,b)×(c,d), can be expressed by the relation z = x 2 +y 2 . The function f(x,y) = x 2 +y 2 is the argument we pass to Plot3D in order to draw the graph of this set of points. We now consider examples of surfaces that cannot be readily expressed as z = f(x,y). Many familiar geometric surfaces, among them cylinders, spheres, and tori, are examples of such surfaces. Let us begin with the example of a sphere of radius 1, whose equation is given by x^2+y^2+z^2=1. It is easy to see that () is equivalent to z=. We can now use the two functions in () with Plot3D and combine the resulting surfaces with Show . One complication arises because of the domain in (). When we try Mathematica complains about the complex numbers associated with certain values in the domain (such as x = y = 1) but still produces a relatively reasonable graph of part of the sphere.

              Equations () describe the surface of the sphere in rectangular coordinates, a coordinate system that is not natural or convenient for plotting the sphere. Instead, we look for the description of this surface in spherical coordinates. In this new coordinate system a point whose rectangular coordinates are x, y, and z is represented by (r, u,v) where x= r u v, y= r u v, z= r u, where u and v are the standard spherical angles. After substituting () into (), we note that the equation for the sphere takes the simple form r = 1. Equations () and (), when combined with ParametricPlot3D , produce the plot of the unit sphere: The output of the above line is shown in Figure . Figure Figure 5: The graph of the sphere of radius 1.

              Many of the familiar shapes and surfaces that we study in mathematics are already programmed in Mathematica and are available as built-in functions. To access them we must first enter and read the special library of shapes into our session of Mathematica . To get, for example, the graph of a torus, we enter the command while draws the surface of the Moebius Strip with inner radius 1 and the outer radius 2 using 160 polygons. is capable of rendering animation. In many applications it is possible, and often desirable, to produce a sequence of graphs and put them in motion. For example, consider the snapshots of a ``vibrating string" defined by

              u(x, t) = sinx cost,
              with x Î (0, p ). For each fixed t, the graph of the function u(·, t) is a snapshot of the string. Let us first generate a sequence of these snapshots: The PlotRange option plots all the snapshots over the same vertical range. The period of vibration is 2 p (the period of cost) hence, we let t vary from 0 to 2 p , at increments of 0.25. When using Mathematica 's Notebook version in Windows 3.1 or on a SUN workstation, we animate the above sequence of graphs by selecting first the cell that contains all of the graphs next we choose Cell from the top menu, from which we select Animate Selected Graphics .

              1. Draw the graph of the following curves in the specified domain.
                1. r (t) = á t, t, t ñ t Î (0,1)
                2. r (t) = á fract12, fract4, frac12+sint ñ -2 p< t < 2 p
                3. r (t) = á e -fract4 sin3t, e -fract4 cos3t, fract12 ñ t Î (0, 4 p ) (use the PlotPoints option of ParametricPlot3D to get a graph with better resolution)
                4. r (t) = á sint, cost, frac1[ Ö (t 2 +1)] ñ t Î (0, 2 p )
                5. r (t) = á sinhfract6, sin(4t), coshfract6 ñ 0 < t < 4 p
                6. r (t) = á t+sin3t, frac1t 2 +1 ñ t Î (-2 p , 2 p )
                1. A circle of radius 2 centered at the origin and located in the xy-plane
                2. A circle of radius 2 centered at the origin and located in the z = 1 plane
                3. The ellipse located in the xy- plane centered at the origin with major and minor axes of 3 and 2, respectively
                4. The curve of intersection of x 2 + y 2 = 1 and z = x
                1. z = x 2 + y 2 , with (x,y) Î (-3, 3)×(-3, 3)
                2. z = [ Ö (x 2 + y 2 )], with -3 < x < 3 and -3 < y < 3
                3. z = 3x 2 + 4y 2 , in (-3, 3)×(-3, 3)
                4. z = sin(x 2 + y 2 ), with x Î (- p , p ) and y Î (- p , p )
                5. z = sin([ Ö (x 2 + y 2 )]), in (- p , p )×(- p , p )
                6. z = sin(x 2 + y 2 )cos(x), in (- p , p )×(- p , p )
                7. z = sin(x 2 + y 2 )cos(y), in (- p , p )×(- p , p )
                8. z = fracsin[ Ö (x 2 +y 2 )][ Ö (x 2 +y 2 )], in (- p , p )×(- p , p )
                1. u(x, t) = sin3x cost x Î (0, p )
                2. u(x, t) = 2 sin3 x cost - 3 sinx cos2t x Î (0, p )
                3. u(x, y, t) = sin3 p x sin2 p y cos p t (x, y) Î (0,1)×(0,1)
                4. The graph of a sequence of spheres whose radii at time t are described by cos2 p t

                1.9 Solving Differential Equations

                has two internal functions, DSolve and NDSolve , that are capable of solving special classes of ordinary differential equations. DSolve is primarily used to find the exact solution to first-order (nonlinear) equations or linear equations with constant coefficient equations. Here are some examples. Consider the initial value problem
                v' + v 2 = 0, v(0) = 1.
                To find a solution to this equation, we enter The reason for the label a will become clear shortly. The output is This output is interpreted as a replacement rule, that is, a rule that replaces v[t] with whenever a is called upon. For example, results in while plots the graph of v . If more than one solution is stored in a , Plot displays the graph of all solutions.

                It is often convenient to define a function v using a : we can now manipulate the function v just like any other function. Assuming v represents the velocity of a particle of unit mass, we determine its kinetic energy during the time interval t Î (0, 3) by Similarly, the acceleration of the particle is obtained by differentiating v[t] once : The inclusion of the independent variable t in v[t] is not optional. For example, does not lead to the correct solution of this equation. Also, the operand cannot be replaced by . Here v' + v 2 = 0 is an equation and not an assignment hence, it is necessary to use . Next, let us consider the differential equation

                m v' + k v = - mg, v(0) = 0.
                Here v represents the velocity of an object of mass m falling under the action of gravity and being resisted by a linear frictional force. From we receive Assuming k and m are positive, it is clear from the above expression that the terminal (limiting) velocity of the object is -fracm gk. With m = 70, k = 10, and g = 9.8, we define and determine v 's terminal velocity by Mathematica returns DSolve is an effective tool for solving linear higher order differential equations with constant coefficients. Consider the initial value problem
                x'' + 3 x' + 2 x = 0, x(0) = 1, x'(0) = 0.
                The exact solution of this problem is found by whose output is The command DSolve works equally well with systems of equations. Consider the initial value problem
                x' = 2 x + 3y, y' = x, x(0) = -2, y(0) = 2.
                We input into Mathematica and obtain the output In the applications where systems of equations appear, the solution pair (x(t), y(t)) is often the parametrization of the path of a fluid particle. To get the graph of this path we enter Methamatica first gives the warning and then proceeds to graph the particle path. To work with compiled functions, first we apply the Evaluate command to the solution pair x[t], y[t] and then plot the result Alternatively, we define a function f as the solution pair x[t], y[t] and then plot f : The second command in Mathematica that is capable of determining solutions of differential equations is NDSolve . This program uses a numerical algorithm (based on the standard Runge-Kutta scheme) and solves linear as well as nonlinear systems of differential equations. Consider the forced nonlinear pendulum equation x'' + 0.1 x' + x = 0.02 t, with initial conditions
                x(0) = 0, x'(0) = 1.
                We have the option of giving () as a second-order equation to Mathematica or as a first-order system. In the first case the syntax is Mathematica responds with which states that it has successfully obtained an approximate solution to the above differential equation and has interpolated a curve through the data points (t i , x i ), where t i Î (0,5) are the discretized values chosen by the Runge-Kutta algorithm. We now define a function x by To get the graph of the solution x of (), we enter Its value at a point such as t = 2.15 is evaluated by which results in 0.8674379209871004. The pair (x(t), x'(t)) is plotted by The last statement brings up the point of solving the second-order differential equation () as a first-order system so that information about x and x' is available simultaneously. To reduce this equation to a first-order system, let x' = y, form which we deduce that y' = - 0.1 y - sinx + 0.02 cost. Thus, () is equivalent to x'=y, y' = - 0.1 y - x + 0.02 t, x(0)=0, y(0)=1. We are now able to determine the approximate solution of () and plot its particle path in the following manner: We first receive the output The particle path is then graphed, as shown in Figure . Figure Figure 6: The phase plane diagram of the solution to x'' + 0.1 x + sinx = 0.02cost.

                1. Use DSolve to solve the following differential equations.
                  1. x' + 3 x = 0
                  2. x' + t x = 3e -t2 , x(1) = 2
                  3. x' + x 3 = 0
                  4. y'' + y = 0, y(0) = 0, y'(0) = 1
                  5. x''' + x' + x = 0
                  6. y'' + y = 0, y(0) = 1, y(1) = -1
                  7. x'' + x = sin2 t
                  8. y'' + y = sint
                  1. x'' + x = 0 x(0) = 0, x'(0) = 1
                  2. y'' +0.1 y'+ siny = 0 y(0) = 0, y'(0) = 1
                  3. y'' +0.1 y' +siny = 0 y(0) = 0, y'(0) = 3
                  4. x'' + 0.1 x + sinx = 0.02 cost x(0) = 0, x'(0) = 1, t Î (0, 100)
                  1. x' = y, y' = -x
                  2. x' = y, y' = -x-0.1 y
                  3. x' = y, y' = -x-y
                  4. x' = y, y' = -x- y 2 .

                  1.10 Vectors, Matrices, and Lists

                  In Mathematica vectors and matrices are entered as lists. For example, the vector a = á -2, 1,3 ñ is entered as Similarly, the matrix
                  B = é
                  is entered as We can write B in matrix form by invoking the MatrixForm command. Thus, gives Elements of a list are accessed by putting the subscript of the element between the delimiters [[. ]] . For example, the first entry of a is a[[1]] while the (1,2)-entry of B is B[[1,2]] . Also, B[[1]] returns which is the first row of B .

                  The length of a list is the number of elements in the list. For example, the vector a and the matrix B both have lengths equal to 3 as can be checked from Length[a] and Length[B] .

                  The standard matrix multiplication is carried out in Mathematica by placing a period ( . ) between the matrices. Thus, to compute the product A 1 A 2 of the matrices

                  A 1 = é
                  , A 2 = é
                  ù ú ú ú û we enter Now results in the 3×3 matrix Similarly, to determine the product A 2 A 1 we enter which results in the 2×2 matrix Mathematica will return an error message if the dimensions of the matrices being multiplied are not compatible. For example, returns Matrix and vector multiplication are carried out in the same manner. If c is the column vector
                  c = é
                  then, after defining it as we compute the product of B (defined earlier) and c by which results in Also the product of the column vector c T with B is determined by whose output is We note that the vector c could also have been defined as a 3×1 matrix: Now B . c returns while c . B returns the error message On the other hand, Transpose[c] . B returns There is a second way of multiplying lists in Mathematica , using the * operand. This operation between two lists A and B of equal lengths returns a list whose entries are the product of individual entries of A and B. For example, returns Here both A and B have length 2. On the other hand, the product returns The Append command appends information to the end of a list. For example, returns the 4×3 matrix with B as its first three rows and á 1,1,1 ñ as its fourth row. This command is particularly useful when, in the course of a calculation, new entries are being computed and this information must be added to a variable and stored for later processing.

                  The commands Det , Inverse , Eigenvalues , and Eigenvectors operate on lists, when applicable, with the standard mathematical results that their names suggest. For example, consider the matrix f

                  f(a) = é
                  Define this matrix in Mathematica by Now returns while leads to Similarly, Eigenvalues[f[1]] and Eigenvectors[f[2]] return the appropriate outputs, respectively. The command Eigensystem combines the outputs of Eigenvalues and Eigenvectors .

                  We often need to plot a set of ordered pairs of numbers. The command ListPlot is the appropriate tool for this task. For example, consider the following four ordered pairs:

                  (1, 0.1), (2, 0.2), (-1, 0.3), (-2, 0.4).
                  To plot them, first we define a list that contains the four pairs and then apply ListPlot to it: The output is shown in Figure . Figure Figure 7: The output of ListPlot . The above discussion touches on only a small portion of what is available in Mathematica concerning lists, matrices, and linear algebra. The reader is encouraged to consult [1] and [2] for more detail on this subject.

                  1.10.1 Lists and Differential Equations

                  1. Let A and B be defined by
                    A =é
                    , B =é
                    ù ú ú ú û . Compute A+B, A-B, AB, BA, 6A, and 3A+2B.
                  2. Consider the matrix
                    A =é
                    where a, b, and c are constants. Find all values of these parameters for which the determinant of A vanishes.
                  3. Let A be the matrix
                    A =é
                    Find all values of a and b for which the determinant of A vanishes.
                  4. Let A be the matrix
                    A =é
                    1. Compute A 5 and A 10 (use MatrixPower ).
                    2. Use Exp and MatrixExp with A. Why are the results different?
                    1. k 1 = k 2 = 10, k 3 = 20 x 1 (0) = 0, x 2 (0) = -1,x 3 (0) = 1, x 1 '(0) = 0, x 2 '(0) = 0, x 3 '(0) = 0 t Î (0, 3)
                    2. k 1 = k 2 = k 3 = k 4 = 10 x 1 (0) = 0, x 2 (0) = 0,x 3 (0) = 0, x 4 (0) = 1, x 1 '(0) = 0, x 2 '(0) = 0, x 3 '(0) = 0, x 4 '(0) = 0 t Î (0, 5)

                    1.11 The , := , , - > , /. Operators

                    1. Determine the outcome of the following statements.
                      1. t == 3
                      2. t = 3 f[t] = Sin[t] Cos[t] f[Pi]
                      3. t = 3 f[t] := Sin[t] Cos[t] f[Pi]
                      4. Sin[a t + b] /. a - > Pi /. b - > 0 /. t - > 1/2
                      5. Solve[x^ 2 - a == 0 /. a - > 3, x]
                      6. Solve[x^ 2 - a == 0, x ] /. a - > 3
                      7. D[Sin[t] /. t - > 3, t]
                      8. D[sin[t], t] /. t - > 3
                      1. Dsolve[x^ 2 - 4 = 0, x]
                      2. Integrate[f[t] = t^ 2, t, 0, 1]
                      3. D[Sin[t], t /. t - > 3]
                      4. DSolve[x'[t] + x[t] == -1, x[0] = 2, x[t], t]

                      1.12 Loops and the Do Command

                      In many numerical applications we need to perform an operation repeatedly while a few parameters may change with each iteration. The Do command in Mathematica is the right tool for such a task. As a first example, consider the sum
                      S = 100
                      i = 1
                      frac1i 2 .
                      One can find an approximate value for S using Do via Mathematica returns 1.63498 . (Try the last program with S = 1 replacing the first line. How are the outputs different?) We also get the same result from Sum : A different context in which Do is useful is in carrying out the type of iterations that arise naturally in the discretization of differential equations. A simple example of this type of application appears in the numerical scheme that produces the Euler approximation of the solution of a first-order differential equation. Consider the first-order differential equation
                      fracdydx = f(x, y), y(x 0 ) = y 0 .
                      The Euler approximation of the solution y(x) seeks a sequence (x n , y n ) that satisfies the difference equation
                      y n+1 = y n + h f(x n , y n ),
                      with x n+1 = x n +h , where h is a fixed small positive number. The following program shows how to generate this sequence and plot the approximate solution using ListPlot . This program is written for f(x, y) = - y + sinx, x 0 = 0, y 0 = 1, h = 0.01, and n = 10. The function of Do in the above program is to perform y = y + h*f[x, y] repeatedly, while updating the value of x and appending the result of x and y to output .
                      1. Use Do and sum the following series.
                        1. å i = 0 20 i
                        2. å i = 1 10 frac1i 2
                        3. å i = 1 100 frac1i 2 . First sum the series using exact arithmetic and then using floating point arithmetic (that is, use the decimal representation of frac1i 2 ).
                        4. å i = 1 1000 frac1i. What is the exact value of the sum? Find its forty decimal point approximation (use N[number, 40] ). Compare this value with the value of the sum when the decimal representation of frac1i is used.
                        1. f(x) = sin2 x x 0 = frac12
                        2. f(x) = sin2 x x 0 = frac32
                        3. f(x) = - sin2 x x 0 = frac12
                        4. f(x) = Ö x + 1 x 0 = 1
                        5. f(x) = sin Ö x + 1 x 0 = 1
                        6. f(x) = frac1x 2 +1 x 0 = 1. Compare the result with the output of
                        7. f(x) = ln2x x 0 = frac12

                        1.13 Examples of Programming in Mathematica

                        A useful feature of Mathematica is that it allows one to input lines of commands from an external file. Using this feature and combining a series of internal functions, we can construct new functions that are specifically customized for certain objectives. We give an example of such a ``program" in the context of differential equations. Its task is to solve a system of differential equations and plot the solution to an initial value problem.

                        Let us consider the system of differential equations = f(x, y, t), = g(x, y, t) subject to the initial data x(0) = x0, y(0) = y0, where f(x, y, t) = x - y + t, g(x, y, t) = x + y + t and x0 = 0.1, y0 = 1.2. We wish to plot the solution of this system over the interval (0, 3). The following lines are saved in a file called ode.m : : When using a word processing software (such as WordPerfect ) to create files for use in Mathematica , it is a good habit to save the files as text-only. Then, after initiating Mathematica , we input ode.m by entering Clearly, if we intend to solve a different set of differential equations we only need to alter the lines in ode.m that define f and g and input the new ode.m program to Mathematica .

                        1. Use an editor and create the files ode.m , odesolver.m , and myode.m . Study the logic of each program carefully. Run these programs separately in Mathematica , and generate Figures and .
                        2. Use ode.m to plot the trajectories of the following systems of differential equations.
                          1. x' = y, y' = -x x(0) = 2, y(0) = -3 0 < t < 4
                          2. x' = 2x-y, y' = x+y x(0) = -1, y(0) = 1 0 < t < 3
                          3. x' = fracy[ Ö (x 2 +y 2 )], y' = -fracx[ Ö (x 2 +y 2 )] x(0) = -2, y(0) = 2 0 < t < 4
                          4. x' = e -t y, y' = -x+e -3t x(0) = 1, y(0) = -1 0 < t < 1.
                          1. x' = y, y' = -0.1 y - siny
                          2. x' = y, y' = - y - y 3
                          3. x' = x + y(1 - x 2 - y 2 ) , y' = y - x(1-x 2 - y 2 )

                          1.14 Glossary of Useful Commands

                          1. PSPrint This command is used primarily in the stand-alone version of Mathematica . The entry PSPrint[Out[x]] , where x marks the output you wish to print, creates a postscript hardcopy of the graphics created and stored in Out[x] . In the Notebook interface version of Mathematica , the Print command in the menu performs this task.
                          2. Table The Table command creates a list of objects. Its syntax is where a list of copies of expr are generated as i and j vary from 1 to n and m , respectively. For example, returns
                          3. Plot, Plot3D, ParametricPlot , and ParametricPlot3D The commands Plot , Plot3D , ParametricPlot , and ParametricPlot3D draw graphs of various two- and three-dimensional representations of functions. For example, draws the graph of f(x) = sinx over the interval (0, 2 p ), while accomplishes the same task using a typical parametrization of the same curve. The commands Plot3D and ParametricPlot3D have similar syntax: and yield the same results.
                          4. Solve and FindRoot and FindRoot find solutions of equations. Typical examples are and
                          5. DSolve and NDSolve and NDSolve find solutions to ordinary differential equations. Their syntax follows the pattern and For example, and
                          6. LaplaceTransform and InverseLaplaceTransform The package that enables Mathematica to compute the Laplace transform of a function is Calculus`LaplaceTransform` . It should be entered in Mathematica at the beginning of a session as Typical commands for computing the Laplace transform and inverse transforms of functions are and

                          [2] Blachman, Nancy, Mathematica: A Practical Approach , Prentice-Hall, Englewood Cliffs, NJ, 1992.


                          The following main topics are contained in the course:

                          1.1. First order differential equations and mathematical models.
                          1.2. Slope fields and initial value problems.
                          1.3. Euler's approximation.
                          1.4. Existence and uniqueness, Picard-Lindelöf theorem (as application of fixed point theorem).
                          1.5. Gronwall's Lemma and the convergence of Euler's method.
                          1.6. Analytic tools: integrating factors, separation of variables, and exact equations.
                          Systems of first order linear differential equations, and linear higher
                          order differential equations: fundamental solutions, the solution
                          2.2. The Wronskian, Abel's theorem.
                          2.3. Analytic tools: undetermined coefficients and the variation of parameters.
                          3. Numerical methods: (embedded) Runge-Kutta methods and adaptivity.
                          4. Stiffness, implicit methods, A-stability.
                          5.1. Introduction to Ito-SDEs: Ito integral, Ito process, Ito formula.
                          5.2 Numerical methods for SDEs: Euler-Maruyama and Milstein methods, weak and strong convergence.