# 1.3 Separable Equations - Mathematics

When a differential equation is of the form (y' = f(x)), we can just integrate: (y = int f(x) dx + C). Unfortunately this method no longer works for the general form of the equation (y' = f(x, y)). Integrating both sides yields

[y = int f(x, y) dx + C]

Notice the dependence on (y) in the integral.

### 1.3.1 Separable equations

Let us suppose that the equation is separable. That is, let us consider

[y' = f(x)g(y),]

for some functions (f(x)) and (g(y)). Let us write the equation in the Leibniz notation

[frac{dy}{dx} = f(x)g(y)]

Then we rewrite the equation as

[frac{dy}{g(y)} = f(x) dx]

Now both sides look like something we can integrate. We obtain

[int frac{dy}{g(y)} = int f(x) dx + C]

If we can find closed form expressions for these two integrals, we can, perhaps, solve for (y.)

Example (PageIndex{1}):

Take the equation

[ y' = xy]

First note that (y = 0) is a solution, so assume (y e 0) from now on. Write the equation as (frac{dy}{dx} = xy,) then

[int frac{dy}{y} = int x dx + C.]

We compute the antiderivatives to get

[ln left vert y ight vert = frac{x^2}{2} + C]

Or

[left vert y ight vert = e^{frac{x^2}{2}} e^{C} = De^{frac{x^2}{2}} ]

where (D > 0) is some constant. Because (y = 0) is a solution and because of the absolute value we actually can write:

(y = De^{frac{x^2}{2}} )

for any number (D) (including zero or negative).

We check:

[y' = Dxe^{frac{x^2}{2}} = x left ( De^{frac{x^2}{2}} ight ) = xy]

We should be a little bit more careful with this method. You may be worried that we were integrating in two different variables. We seemed to be doing a different operation to each side. Let us work this method out more rigorously.

[frac{dy}{dx} = f(x)g(y)]

We rewrite the equation as follows. Note that (y = y(x)) is a function of (x) and so is (frac{dy}{dx}!)

[frac{1}{g(y)} frac{dy}{dx} = f(x)]

We integrate both sides with respect to (x.)

[int frac{1}{g(y)} frac{dy}{dx} dx = int f(x) dx + C]

We can use the change of variables formula.

[int frac{1}{g(y)} dy = int f(x) dx + C]

And we are done.

It is clear that we might sometimes get stuck even if we can do the integration. For example, take the separable equation

[y' = frac{xy}{y^2 + 1}]

We separate variables,

[frac{y^2 + 1}{y} dy = left ( y + frac{1}{y} ight ) dy = x dx]

We integrate to get

[frac{y^2}{2} + ln left vert y ight vert = frac{x^2}{2} + C]

or perhaps the easier looking expression (where (D = 2C))

[y^2 + 2ln left vert y ight vert = x^2 + D]

It is not easy to find the solution explicitly as it is hard to solve for (y). We, therefore, leave the solution in this form and call it an implicit solution. It is still easy to check that an implicit solution satisfies the differential equation. In this case, we differentiate to get

[y' left ( 2y + frac{2}{y} ight ) = 2x]

It is simple to see that the differential equation holds. If you want to compute values for (y), you might have to be tricky. For example, you can graph (x) as a function of (y), and then flip your paper. Computers are also good at some of these tricks.

We note that the above equation also has the solution (y = 0). The general solution is (y^2 + 2ln left vert y ight vert = x^2 + C) together with (y = 0). These outlying solutions such as (y = 0) are sometimes called singular solutions.

Example (PageIndex{2}):

Solve (x^2y' = 1 - x^2 + y^2 -x^2y^2), (y(1) = 0.)

First factor the right hand side to obtain

[x^2y' = left ( 1- x^2 ight ) left ( 1 + y^2 ight )]

We separate variables, integrate and solve for (y)

[frac{y'}{1 + y^2} = frac {1 - x^2}{x^2},]

[frac{y'}{1 + y^2} = frac {1}{ x^2} -1,]

[ ext{arctan}(y) = -frac{1}{x^2} - x + C,]

[y = an left( -frac{1}{x} - x + C ight )]

Now solve for the initial condition, (0 = an (-2 + C)) to get ( C = 2 ( { m{~or~}} 2 + pi, { m{~etc~}} dots)). The solution we are seeking is, therefore,

[y = an left( -frac{1}{x} - x + C ight )]

Example (PageIndex{3}):

Bob made a cup of coffee, and Bob likes to drink coffee only once it will not burn him at 60 degrees. Initially at time (t = 0) minutes, Bob measured the temperature and the coffee was 89 degrees Celsius. One minute later, Bob measured the coffee again and it had 85 degrees. The temperature of the room (the ambient temperature) is 22 degrees. When should Bob start drinking?

Let (T) be the temperature of the coffee, and let (A) be the ambient (room) temperature. Newton’s law of cooling states that the rate at which the temperature of the coffee is changing is proportional to the difference between the ambient temperature and the temperature of the coffee. That is,

[frac{dT}{dt} = k(A - T),]

for some constant (k). For our setup ( A = 22), (T(0) = 89), (T(1) = 85). We separate variables and integrate (let (C) and (D) denote arbitrary constants)

[frac{1}{T -A} frac {dT}{dt} = -k,]

[ln (T - A) = -kt + C, , , , , , left ( ext {note that} T - A > 0 ight )]

[ T - A = De^{-kt},]

[ T = A + De^{-kt}]

That is, ( T = 22 + De^{-kt}). We plug in the first condition: ( 89 = T(0) = 22 + D), and hence (D = 67). So (T = 22 + 67e^{-kt}). The second condition says ( 85 = T(1) = 22 + 67e^{-k}). Solving for (k) we get (k = -ln frac{85-22}{67} approx 0.0616). Now we solve for the time (t) that gives us a temperature of 60 degrees. That is, we solve (60 = 22 + 67e^{-0.0616t}) to get (t = -frac {ln frac {60 -22}{67}}{0.0616} approx 9.21) minutes. So Bob can begin to drink the coffee at just over 9 minutes from the time Bob made it. That is probably about the amount of time it took us to calculate how long it would take.

Example (PageIndex{4}):

Find the general solution to (y' = -frac {xy^2}{3}) (including singular solutions).

First note that ( y = 0) is a solution (a singular solution). So assume that ( y e 0) and write

[ -frac {3}{y^2} y' = x ,]

[ frac {3}{y} = frac {x^2}{2} + C,]

[ y = frac {3}{ frac{x^2}{2} + C} = frac {6}{x^2 + 2C}.]

## Separable Algebras

This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups.

The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.

Graduate students and researchers interested in algebra.

#### Reviews & Endorsements

The book is neatly arranged. It can be used as a textbook for self-study and as a reference text for most of the topics related to separability. It will be a valuable resource for students and researchers and has the potential to be a standard reference on separable algebras for many years.

-- Wolfgang Rump, Mathematical Reviews

The thorough and comprehensive treatment of separable, Azumaya, and tale algebras, Hensel rings, the Galois theory of rings, and Galois cohomology of rings makes the book under review an indispensable reference for the graduate student interested in these topics. As an added bonus, the book comes with a rich, 155 item, bibliography, well-chosen examples, calculations, and sets of exercises in each chapter, which makes this book an excellent textbook for self-study or for a topics course on separable algebras.

-- Felipe Zaldivar, MAA Reviews

## 1.3 Separable Equations - Mathematics

A separable first-order ode has the form:

where g(t) and h(y) are given functions. Note that y'(t) is the product of functions of the independent variable and dependent variable.

To solve this problem, divide by h(y):

Here we have relabeled 1/h(y(t)) by H(y(t)). Now integrate both sides with respect to t

The left-hand side can be simplified using the method of substition for integrals. Let u=y(t). Then du=y'(t)dt. We then have

The final answer is (we change from the dummy variable u to the dummy variable y)

We can also write the answer in terms of definite integrals if an initial condition is given. We have

Make the subsitution in the first integral u=y(s), du=y'(s)ds. The lower limit of the new integral is

The upper limit of the new integral is y(t). We then have

In this case H(y)=y and g(t)=exp(t)+1. Computing the integrals we have

The general solution of the ode is

Applying the initial condition, we find that

C=9/2-exp(1)-1=0.782. The solution of the ode is

We can solve for y. We obtain

Notice that we take the positive square root. Why? We know that y(1)=3, a positive number.

## Uncorrelation without mean-independence

A simple counterexample is ((X,Y)) uniformly distributed on the vertices of a regular polygon centered on the origin, not symmetric with respect to either axis.

For example, let ((X, Y)) have uniform distribution with values in [ig<(1,3), (-3,1), (-1,-3), (3,-1)ig>.]

Then (mathbb(XY) = 0) and (mathbb(X)=mathbb(Y)=0), so (X) and (Y) are uncorrelated.

Yet (mathbb(X|Y=1) = -3), (mathbb(X|Y=3)=1) so we don’t have (mathbb(X|Y) = mathbb(X)). Likewise, we don’t have (mathbb(Y|X) = mathbb(Y)).

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Credit Hours: 3

Description: Study of classical and modern theories about functions having some integral expression which is maximal, minimal, or critical. Geodesics, brachistochrone problem, minimal surfaces, and numerous applications to physics. Euler-Lagrange equations, 1st and 2nd variations, Hamilton's Principle.

MATH 412 - PROBABILITY THEORY

Short Title: PROBABILITY THEORY

Department: Mathematics

Course Type: Lecture

Distribution Group: Distribution Group III

Credit Hours: 3

Prerequisite(s): MATH 321 or MATH 331

Description: A simultaneous introduction to probability theory and measure theory, from basic definitions to the central limit theorem. The selection of topics in measure theory is in the service of probability theory, and the course carefully examines interplay between the analytic and probabilistic notions.

MATH 423 - PARTIAL DIFFERENTIAL EQUATIONS I

Short Title: PARTIAL DIFFERENTIAL EQNS I

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Description: First order of partial differential equations. The method of characteristics. Analysis of the solutions of the wave equation, heat equation and Laplace's equation. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Cross-list: CAAM 423. Graduate/Undergraduate Equivalency: MATH 513. Recommended Prerequisite(s): MATH 321 AND MATH 322 Mutually Exclusive: Cannot register for MATH 423 if student has credit for MATH 513.

MATH 424 - TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS

Short Title: TOPICS IN PARTIAL DIFF EQNS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 423

Description: Continuation of MATH 423. Analysis of the solutions of second order differential equations. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Euler and Navier-Stokes equations. Graduate/Undergraduate Equivalency: MATH 514. Mutually Exclusive: Cannot register for MATH 424 if student has credit for MATH 514. Repeatable for Credit.

MATH 425 - INTEGRATION THEORY

Short Title: INTEGRATION THEORY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 321 or MATH 331

Description: Lebesgue theory of measure and integration. Graduate/Undergraduate Equivalency: MATH 515. Mutually Exclusive: Cannot register for MATH 425 if student has credit for MATH 515.

MATH 426 - TOPICS IN REAL ANALYSIS

Short Title: TOPICS IN REAL ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 425

Description: Content varies from year to year. May include Fourier series, harmonic analysis, probability theory, advanced topics in measure theory, ergodic theory, and elliptic integrals. Graduate/Undergraduate Equivalency: MATH 516. Mutually Exclusive: Cannot register for MATH 426 if student has credit for MATH 516. Repeatable for Credit.

MATH 427 - COMPLEX ANALYSIS

Short Title: COMPLEX ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 354 or MATH 222 or MATH 302

Description: Study of the Cauchy-Riemann equation, power series, Cauchy's integral formula, residue calculus, and conformal mappings. Emphasis on the theory. Graduate/Undergraduate Equivalency: MATH 517. Recommended Prerequisite(s): MATH 321 or MATH 331. Mutually Exclusive: Cannot register for MATH 427 if student has credit for MATH 382/MATH 517.

MATH 428 - TOPICS IN COMPLEX ANALYSIS

Short Title: TOPICS IN COMPLEX ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 382 or MATH 427

Description: Special topics include Riemann mapping theorem, Runge's Theorem, elliptic function theory, prime number theorem, Riemann surfaces, et al. Graduate/Undergraduate Equivalency: MATH 518. Mutually Exclusive: Cannot register for MATH 428 if student has credit for MATH 518. Repeatable for Credit.

MATH 435 - DYNAMICAL SYSTEMS

Short Title: DYNAMICAL SYSTEMS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Description: Existence and uniqueness for solutions of ordinary differential equations and difference equations, linear systems, nonlinear systems, stability, periodic solutions, bifurcation theory. Theory and theoretical examples are complemented by computational, model driven examples from biological and physical sciences. Cross-list: CAAM 435. Recommended Prerequisite(s): (MATH 212 or MATH 221) and (CAAM 335 or MATH 355 or MATH 354) and (MATH 302 or MATH 321 or MATH 331)

MATH 443 - GENERAL TOPOLOGY

Short Title: GENERAL TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 321 or MATH 331

Description: Study of basic point set topology. Includes a treatment of cardinality and well ordering, as well as metrization. Graduate/Undergraduate Equivalency: MATH 538. Mutually Exclusive: Cannot register for MATH 443 if student has credit for MATH 538.

MATH 444 - GEOMETRIC TOPOLOGY

Short Title: GEOMETRIC TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 356 and MATH 443 and (MATH 322 or MATH 370 or MATH 401)

Description: Introduction to algebraic methods in topology and differential topology. Elementary homotopy theory. Covering spaces. Differentiable Manifolds. Graduate/Undergraduate Equivalency: MATH 539. Mutually Exclusive: Cannot register for MATH 444 if student has credit for MATH 539.

MATH 445 - ALGEBRAIC TOPOLOGY

Short Title: ALGEBRAIC TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 444

Description: Introduction to the theory of homology. Includes simplicial complexes, cell complexes and cellular homology and cohomology, as well as manifolds, and Poincare duality. Graduate/Undergraduate Equivalency: MATH 540. Mutually Exclusive: Cannot register for MATH 445 if student has credit for MATH 540.

MATH 448 - CONCRETE MATHEMATICS

Short Title: CONCRETE MATHEMATICS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): COMP 182 or MATH 220 or MATH 221 or MATH 302 or MATH 354

Description: Concrete mathematics is a blend of continuous and discrete mathematics. Major topics include sums, recurrences, integer functions, elementary number theory, binomial coefficients, generating functions, discrete probability and asymptotic methods. Cross-list: COMP 448.

MATH 463 - ADVANCED ALGEBRA I

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 357

Description: A first graduate course in Algebra. Advanced topics in group theory: semi-direct products, dual groups, inverse limits and completions. Galois Theory, including infinite Galois xtensions. Advanced topics in ring theory: radicals, local rings, primary decomposition, the prime spectrum of a ring. Groebner Bases elimination theory. Modules: the Cayley-Hamilton Theorem and Nakayama's lemma localization and local properties composition series and the Jordan-Hölder theorem structure theorem for finitely generated modules over a PID. Tensor products: universal properties, multilinear algebra. Introduction to categories and functors. Graduate/Undergraduate Equivalency: MATH 563. Mutually Exclusive: Cannot register for MATH 463 if student has credit for MATH 563.

MATH 464 - ADVANCED ALGEBRA II

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 463

Description: Continuation of MATH 463. Tensor and exterior algebra, introductory commutative algebra, structure of modules, and elements of homological algebra. Additional advanced topics may include representations of finite groups and affine algebraic geometry. Graduate/Undergraduate Equivalency: MATH 564. Mutually Exclusive: Cannot register for MATH 464 if student has credit for MATH 564.

MATH 465 - TOPICS IN ALGEBRA: INTRODUCTION TO ALGEBRAIC GEOMETRY

Short Title: TOPICS IN ALGEBRA

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Description: Varieties as solution sets of systems of polynomial equations, varieties in projective space, rational and regular functions, maps of varieties, local properties and singularities. Graduate/Undergraduate Equivalency: MATH 565. Mutually Exclusive: Cannot register for MATH 465 if student has credit for MATH 565. Repeatable for Credit.

MATH 466 - TOPICS IN ALGEBRA II

Short Title: TOPICS IN ALGEBRA II

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Description: Content varies from year to year. Graduate/Undergraduate Equivalency: MATH 566. Mutually Exclusive: Cannot register for MATH 466 if student has credit for MATH 566.

MATH 468 - POTPOURRI

Short Title: POTPOURRI

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Description: This course deals with miscellaneous special topics not covered in other courses. Repeatable for Credit.

MATH 471 - MATHEMATICS OF APERIODIC ORDER

Short Title: MATHEMATICS OF APERIODIC ORDER

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 321 or MATH 354 or MATH 355

Description: Mathematical models of quasicrystals, whose discovery in the early 1980's led to a paradigm shift in materials science. Topics include: classical theory of ordered structures (i.e., lattices modeling crystals), Delone subsets and tilings of Euclidean space, aperiodically ordered structures generated by inflation or cut-and-project schemes. Graduate/Undergraduate Equivalency: MATH 571. Recommended Prerequisite(s): MATH 356. Mutually Exclusive: Cannot register for MATH 471 if student has credit for MATH 571.

MATH 477 - SPECIAL TOPICS

Short Title: SPECIAL TOPICS

Department: Mathematics

Course Type: Internship/Practicum, Seminar, Lecture, Laboratory

Credit Hours: 1-4

Description: Topics and credit hours may vary each semester. Contact department for current semester's topic(s). Repeatable for Credit.

MATH 479 - MATHEMATICS UNDERGRADUATE RESEARCH

Department: Mathematics

Course Type: Research

Credit Hours: 1-3

Description: In depth investigation of a particular area of mathematics of mutual interest to the student and the faculty adviser. Instructor Permission Required. Repeatable for Credit.

Department: Mathematics

Course Type: Independent Study

Credit Hours: 1-6

Description: Repeatable for Credit.

MATH 498 - RESEARCH THEMES IN THE MATHEMATICAL SCIENCES

Short Title: RESEARCH THEMES IN MATH. SCI.

Department: Mathematics

Course Type: Seminar

Credit Hours: 1-3

Description: A seminar course that will cover selected theme of general research in the mathematical sciences from the perspectives of mathematics, computational and applied mathematics and statistics. The course may be repeated multiple times for credit. Cross-list: CAAM 498, STAT 498. Graduate/Undergraduate Equivalency: MATH 698. Mutually Exclusive: Cannot register for MATH 498 if student has credit for MATH 698. Repeatable for Credit.

MATH 499 - MATHEMATICAL SCIENCES VIGRE SEMINAR

Short Title: MATHEMATICAL SCIENCES

Department: Mathematics

Course Type: Seminar

Credit Hours: 1-3

Description: Repeatable for Credit.

MATH 500 - DIFFERENTIAL GEOMETRY

Short Title: DIFFERENTIAL GEOMETRY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 444 or MATH 539

Description: A graduate course in smooth and Riemannian manifolds. Tensors, Riemannian metrics, differential forms. Lie derivatives. Distributions and foliations, including the Frobenius Theorem and an introduction to contact structures. Lie groups and the exponential map. Connections on Vector Bundles. Geodesics and completeness. Curvature. First and second variations of length and area. Jacobi Fields. Additional topics may vary from year to year. Graduate/Undergraduate Equivalency: MATH 402. Mutually Exclusive: Cannot register for MATH 500 if student has credit for MATH 402. Repeatable for Credit.

MATH 501 - TOPICS IN DIFFERENTIAL GEOMETRY

Short Title: TOPICS DIFFERENTIAL GEOMETRY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 502 - TOPICS IN DIFFERENTIAL GEOMETRY

Short Title: TOPIC DIFFERENTIAL GEOMETRY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 513 - PARTIAL DIFFERENTIAL EQUATIONS I

Short Title: PARTIAL DIFFERENTIAL EQNS I

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: First order of partial differential equations. The method of characteristics. Analysis of the solutions of the wave equation, heat equation and Laplace's equation. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Additional course work is required beyond the undergraduate course requirements. Cross-list: CAAM 523. Graduate/Undergraduate Equivalency: MATH 423. Recommended Prerequisite(s): MATH 321 AND MATH 322 Mutually Exclusive: Cannot register for MATH 513 if student has credit for MATH 423.

MATH 514 - TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS

Short Title: TOPICS IN PARTIAL DIFF EQNS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 513 or MATH 423

Description: Continuation of MATH 513. Analysis of the solutions of second order differential equations. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Euler and Navier-Stokes equations. Graduate/Undergraduate Equivalency: MATH 424. Mutually Exclusive: Cannot register for MATH 514 if student has credit for MATH 424.

MATH 515 - INTEGRATION THEORY

Short Title: INTEGRATION THEORY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: . Graduate/Undergraduate Equivalency: MATH 425. Mutually Exclusive: Cannot register for MATH 515 if student has credit for MATH 425.

MATH 516 - TOPICS IN REAL ANALYSIS

Short Title: TOPICS IN REAL ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 425

Description: . Graduate/Undergraduate Equivalency: MATH 426. Mutually Exclusive: Cannot register for MATH 516 if student has credit for MATH 426. Repeatable for Credit.

MATH 517 - COMPLEX ANALYSIS

Short Title: COMPLEX ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Graduate/Undergraduate Equivalency: MATH 427. Mutually Exclusive: Cannot register for MATH 517 if student has credit for MATH 382/MATH 427.

MATH 518 - TOPICS IN COMPLEX ANALYSIS

Short Title: TOPICS IN COMPLEX ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 382 or MATH 427

Description: . Graduate/Undergraduate Equivalency: MATH 428. Mutually Exclusive: Cannot register for MATH 518 if student has credit for MATH 428. Repeatable for Credit.

MATH 521 - ADVANCED TOPICS IN REAL ANALYSIS

Short Title: ADV TOPIC: REAL ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 522 - TOPICS IN ANALYSIS

Short Title: TOPICS IN ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 523 - FUNCTIONAL ANALYSIS

Short Title: FUNCTIONAL ANALYSIS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Banach spaces: review of L^p spaces, linear operators, dual space, Hahn-Banach theorem, weak topologies, Banach-Alaoglu theorem, compact and bounded operators, closed graph theorem Hilbert spaces: self-adjoint and unitary operators (including spectral theorem), symmetric operators and self-adjoint extensions if time allows, distributions and Sobolev spaces. Repeatable for Credit.

MATH 524 - TOPICS IN PARTIAL DIFFERENTIAL EQUATIONS

Short Title: TOPICS IN PDE

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 527 - ERGODIC THEORY AND TOPOLOGICAL DYNAMICS

Short Title: ERGODIC THRY&TOP DYNAMICS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 528 - ERGODIC THEORY AND TOPOLOGICAL DYNAMICS

Short Title: ERGODIC THRY&TOPOLOGICAL DYN

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Repeatable for Credit.

MATH 538 - GENERAL TOPOLOGY

Short Title: GENERAL TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: . Graduate/Undergraduate Equivalency: MATH 443. Mutually Exclusive: Cannot register for MATH 538 if student has credit for MATH 443.

MATH 539 - GEOMETRIC TOPOLOGY

Short Title: GEOMETRIC TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 356 and MATH 443

Description: . Graduate/Undergraduate Equivalency: MATH 444. Mutually Exclusive: Cannot register for MATH 539 if student has credit for MATH 444.

MATH 540 - ALGEBRAIC TOPOLOGY

Short Title: ALGEBRAIC TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 539

Description: . Graduate/Undergraduate Equivalency: MATH 445. Mutually Exclusive: Cannot register for MATH 540 if student has credit for MATH 445.

MATH 541 - TOPICS IN TOPOLOGY

Short Title: TOPICS IN TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 542 - TOPICS IN ADVANCED TOPOLOGY

Short Title: TOPICS IN ADVANCED TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Topic to be announced. Repeatable for Credit.

MATH 543 - TOPICS IN LOW-DIMENSIONAL TOPOLOGY

Short Title: TOPICS IN L-D TOPOLOGY

Department: Mathematics

Course Type: Lecture

Credit Hours: 1-3

Restrictions: Enrollment is limited to Graduate level students.

Description: Repeatable for Credit.

MATH 544 - TOPOLOGY OF MANIFOLDS

Short Title: TOPOLOGY OF MANIFOLDS

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): (MATH 444 or MATH 539) and (MATH 445 or MATH 540)

Description: A graduate course on the topology of fiber bundles, especially vector bundles and principal bundles, as well as their characteristic classes. It will cover differential forms as well as Stiefel-Whitney, Euler, Chern, and Pontryagin classes. If time allows, other topics may be included. The prerequisites for the class are the material from Math 444/539 and Math 445/540. In particular, the student should be familiar with smooth manifolds, the tangent spaces, homotopy groups, covering spaces, and homology groups.

MATH 563 - ADVANCED ALGEBRA I

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 357

Description: A first graduate course in Algebra. Advanced topics in group theory: semi-direct products, dual groups, inverse limits and completions. Galois Theory, including infinite Galois Extensions. Advanced topics in ring theory: radicals, local rings, primary decomposition, the prime spectrum of a ring. Groebner Bases elimination theory. Modules: the Cayley-Hamilton Theorem and Nakayama's lemma localization and local properties composition series and the Jordan-Hölder theorem structure theorem for finitely generated modules over a PID. Tensor products: universal properties, multilinear algebra. Introduction to categories and functors. Graduate/Undergraduate Equivalency: MATH 463. Mutually Exclusive: Cannot register for MATH 563 if student has credit for MATH 463.

MATH 564 - ADVANCED ALGEBRA II

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Prerequisite(s): MATH 463 or MATH 563

Description: A continuation of Math 563. Integrality: going up and down theorems. Noether Normalization and Hilbert's Nullstellensatz. Artinian Rings, Discrete Valuation Rings and Dedekind Domains, with applications: rings of integers in number rings, prime decomposition in number ring extensions. Completions: topologies, filtrations. Graded modules and associated graded rings. Homological algebra: complexes, homology, homotopies injective, projective and flat modules. Derived functors: Ext and Tor. Introduction to group cohomology and spectral sequences. Graduate/Undergraduate Equivalency: MATH 464. Mutually Exclusive: Cannot register for MATH 564 if student has credit for MATH 464.

MATH 565 - TOPICS IN ALGEBRA: INTRODUCTION TO ALGEBRAIC GEOMETRY

Short Title: TOPICS IN ALGEBRA

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Varieties as solution sets of systems of polynomial equations, varieties in projective space, rational and regular functions, maps of varieties, local properties and singularities. Graduate/Undergraduate Equivalency: MATH 465. Mutually Exclusive: Cannot register for MATH 565 if student has credit for MATH 465. Repeatable for Credit.

MATH 566 - TOPICS IN ALGEBRA II

Short Title: TOPICS IN ALGEBRA II

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: . Graduate/Undergraduate Equivalency: MATH 466. Mutually Exclusive: Cannot register for MATH 566 if student has credit for MATH 466. Repeatable for Credit.

MATH 567 - TOPICS IN ALGEBRAIC GEOMETRY

Short Title: TOPICS IN ALGEBRAIC GEOMETRY

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Possible topics include rational points on algebraic varieties, moduli spaces, deformation theory, and Hodge structures. Recommended Prerequisite(s): MATH 463 and MATH 464. Repeatable for Credit.

MATH 571 - MATHEMATICS OF APERIODIC ORDER

Short Title: MATHEMATICS OF APERIODIC ORDER

Department: Mathematics

Course Type: Lecture

Credit Hours: 3

Restrictions: Enrollment is limited to Graduate level students.

Description: Mathematical models of quasicrystals, whose discovery in the early 1980's led to a paradigm shift in materials science. Topics include: classical theory of ordered structures (i.e., lattices modeling crystals), Delone subsets and tilings of Euclidean space, aperiodically ordered structures generated by inflation or cut-and-project schemes. Graduate/Undergraduate Equivalency: MATH 471. Recommended Prerequisite(s): MATH 356 Mutually Exclusive: Cannot register for MATH 571 if student has credit for MATH 471.

MATH 590 - CURRENT MATHEMATICS SEMINAR

Short Title: CURRENT MATHEMATICS SEMINAR

Department: Mathematics

Course Type: Seminar

Credit Hour: 1

Restrictions: Enrollment is limited to Graduate level students.

Description: Lectures on topics of recent research in mathematics delivered by mathematics graduate students and faculty. Repeatable for Credit.

MATH 591 - GRADUATE TEACHING SEMINAR

Department: Mathematics

Course Type: Seminar

Credit Hour: 1

Restrictions: Enrollment is limited to Graduate level students.

Description: Discussion on teaching issues and practice lectures by participants as preparation for classroom teaching of mathematics. Repeatable for Credit.

MATH 677 - SPECIAL TOPICS

Short Title: SPECIAL TOPICS

Department: Mathematics

Course Type: Internship/Practicum, Seminar, Lecture, Laboratory

Credit Hours: 1-4

Description: Topics and credit hours vary each semester. Contact department for current semester's topic(s). Repeatable for Credit.

MATH 680 - MATHEMATICS COLLOQUIUM

Short Title: MATHEMATICS COLLOQUIUM

Department: Mathematics

Course Type: Lecture

Credit Hour: 1

Restrictions: Enrollment is limited to Graduate level students.

Description: Presentations of research topics in mathematics and related fields. Repeatable for Credit.

MATH 681 - TOPOLOGY SEMINAR

Short Title: TOPOLOGY SEMINAR

Department: Mathematics

Course Type: Seminar

Credit Hour: 1

Restrictions: Enrollment is limited to Graduate level students.

Description: Presentations of research in topology and related areas. Repeatable for Credit.

MATH 682 - ALGEBRAIC GEOMETRY SEMINAR

Short Title: ALGEBRAIC GEOMETRY SEMINAR

Department: Mathematics

Course Type: Seminar

Credit Hour: 1

Restrictions: Enrollment is limited to Graduate level students.

Description: Presentations of research in algebraic geometry and related areas. Repeatable for Credit.

MATH 683 - GEOMETRY AND ANALYSIS SEMINAR

Short Title: GEOMETRY AND ANALYSIS SEMINAR

Department: Mathematics

Course Type: Seminar

Credit Hour: 1

Restrictions: Enrollment is limited to Graduate level students.

Description: Presentations of research in geometric analysis, mathematical physics, dynamics and related areas. Repeatable for Credit.

Department: Mathematics

Course Type: Independent Study

Credit Hours: 1-6

Restrictions: Enrollment is limited to Graduate level students.

Description: Repeatable for Credit.

MATH 698 - RESEARCH THEMES IN THE MATHEMATICAL SCIENCES

Short Title: RESEARCH THEMES IN MATH. SCI.

Department: Mathematics

Course Type: Seminar

Credit Hours: 1-3

Restrictions: Enrollment is limited to Graduate level students.

Description: A seminar course that will cover selected theme of general research in the mathematical sciences from the perspectives of mathematics, computational and applied mathematics and statistics. The course may be repeated multiple times for credit. Cross-list: CAAM 698, STAT 698. Graduate/Undergraduate Equivalency: MATH 498. Mutually Exclusive: Cannot register for MATH 698 if student has credit for MATH 498. Repeatable for Credit.

MATH 699 - MATHEMATICAL SCIENCES VIGRE SEMINAR

Short Title: MATHEMATICAL SCIENCES

Department: Mathematics

Course Type: Seminar

Credit Hours: 1-9

Restrictions: Enrollment is limited to Graduate level students.

Description: Repeatable for Credit.

MATH 700 - SUMMER RESEARCH FOR PHD STUDENTS

Short Title: SUMMER RESEARCH

Department: Mathematics

Course Type: Research

Credit Hours: 9

Restrictions: Enrollment is limited to Graduate level students. Enrollment limited to students in a Doctor of Philosophy degree.

Description: Summer research for MATH PhD students. Can be repeated for credit. Repeatable for Credit.

MATH 800 - GRADUATE THESIS AND RESEARCH

Short Title: GRADUATE THESIS AND RESEARCH

Department: Mathematics

Course Type: Research

Credit Hours: 1-15

Restrictions: Enrollment is limited to Graduate level students.

## 1.3 Separable Equations - Mathematics

Builds the Affine Cipher Translation Algorithm from a string given an a and b value

This calculator determines the nth automorphic number

Determines the product of two expressions using boolean algebra.

Given a set of modulo equations in the form:
x &equiv a mod b
x &equiv c mod d
x &equiv e mod f

the calculator will use the Chinese Remainder Theorem to find the lowest possible solution for x in each modulus equation.
Given that the ni portions are not pairwise coprime and you entered two modulo equations, then the calculator will attempt to solve using the Method of Successive Subsitution

Takes any natural number using the Collatz Conjecture and reduces it down to 1.

This calculator determines the nth composite number. Helps you generate composite numbers.

Given a possible congruence relation a &equiv b (mod n), this determines if the relation holds true (b is congruent to c modulo n).

Given a set of partitions, this determines the cross partitions.

This calculator determines the nth decagonal number

Calculates the number of derangements/subfactorial !n.

Determines how many (n) digit numbers can be formed based on a variety of criteria.

Solves for ax + by = c using integer solutions if they exist

Given 2 numbers a and b, this calculates the following
1) The Greatest Common Divisor (GCD) using Euclids Algorithm
2) x and y in Bézouts Identity ax + by = d using Euclids Extended Algorithm Extended Euclidean Algorithm

Given a positive integer (n), this calculates Euler's totient, also known as &phi

For any integer a and a prime number p, this demonstrates Fermats Little Theorem.

Demonstrates the addition table and multiplication table for a finite field (Galois Field) of n denoted GF(n).

This calculator generates 5000 iterations for the development of the gamma constant &gamma

Given an original group of certain types of member, this determines how many groups/teams can be formed using a certain condition.

This calculator determines the nth heptagonal number

This calculator determines the nth hexagonal number

Calculates hyperbolic function values: sinh, cosh, tanh, csch, sech, coth

Calculates hyperbolic function values: arcsinh, arccosh, arctanh, arccsch, arcsech, arccoth

Given a set of data, this interpolates using the following methods:
* Linear Interpolation
* Nearest Neighbor (Piecewise Constant)
* Polynomial Interpolation

Given a partitioned interval, this evaluates the norm (mesh) by calculating each subinterval

Builds the Lagrange Theorem Notation (Bachet Conjecture) for any natural number using the Sum of four squares.

Given a word, this determines the number of unique arrangements of letters in the word.

Given an modular equation ax &equiv b (mod m), this solves for x if a solution exists

Using the linear congruential generator algorithm, this generates a list of random numbers based on your inputs

Solves x n mod p using the following methods:
* Modular Exponentiation
* Successive Squaring

Given 2 integers a and b, this modulo calculator determines a mod b or simplifies modular arithmetic such as 7 mod 3 + 5 mod 8 - 32 mod 5

Calculates the multifactorial n! (m)

This calculator determines the nth nonagonal number

This calculator determines if an integer you entered has any of the following properties:
* Even Numbers or Odd Numbers (Parity Function or even-odd numbers)
* Evil Numbers or Odious Numbers
* Perfect Numbers, Abundant Numbers, or Deficient Numbers
* Triangular Numbers
* Prime Numbers or Composite Numbers
* Automorphic (Curious)
* Undulating Numbers
* Square Numbers
* Cube Numbers
* Palindrome Numbers
* Repunit Numbers
* Apocalyptic Power
* Pentagonal
* Tetrahedral (Pyramidal)
* Narcissistic (Plus Perfect)
* Catalan
* Repunit

This calculator determines the nth octagonal number

Given a population size (n) and a group population of (m), this calculator determines how many ordered or unordered groups of (m) can be formed from (n)

This calculator determines the nth pentagonal number

Given a prime number p and a potential root of b, this determines if b is a primitive root of p.

Given 2 positive integers n and d, this displays the quotient remainder theorem.

This calculator determines the nth rectangular number

Using the Sieve of Eratosthenes algorithm, this will show how many prime numbers are less than a number (n).

This calculator determines the nth square number

This calculator determines the nth triangular number. Generates composite numbers.

Sets up a truth table based on a logical statement of 1, 2 or 3 letters with statements such as propositions, equivalence, conjunction, disjunction, negation. Includes modus ponens.

Logic and reasoning, math fact practice and more make up the free educational games at Hooda Math. There are quite a few games that call for higher order thinking and ask students to problem solve to complete the activities. These challenges help to sharpen students’ math skills through an engaging online activity.

Manga High offers free and subscription packages to classroom and homeschool teachers. The free website allows students to play basic games to reinforce math skills and compete against the computer or others. The subscription version gives teachers the opportunity to track students’ progress and determine if there are any gaps in their basic skills.

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## 1.3 Separable Equations - Mathematics

Natasha Glydon

Math is in every kitchen, on every recipe card, and at each holiday gathering. The mathematics of cooking often goes unnoticed, but in reality, there is a large quantity of math skills involved in cooking and baking.

Conversions

Most ranges have dials that display the cooking temperature of the oven. In North America, most of these temperatures are written in Fahrenheit and usually are in increments of 25°. In Canada, recipe and oven temperatures are often presented in degrees Celsius. It is important then to understand how to convert a Fahrenheit temperature to an appropriate Celsius temperature. For example, let&rsquos say your oven displays Fahrenheit temperatures with 50° increments. Your recipe tells you to bake your dish at 220°C. What temperature do you turn your oven to? Well, you will need to convert 220°C to a Fahrenheit measurement.

We use this formula to convert Celsius to Fahrenheit:

To make sure you do not over bake the cookies, you will need to set the oven to 428°F. But remember, your oven only displays the temperature in 50° increments, so you must estimate on the dial where 428°F is, somewhere between 400°F and 450°F. The relationship between celsius and fahrenheit is a linear function:

We also use conversions when we bake or cook to convert sizes and amounts. Many recipes are written in imperial units (teaspoon, tablespoons, and cups). Some newer recipes and measuring devices in Canada are labeled in metric units, such as milliliters (mL). If the recipe calls for ½ cup of butter and your measuring equipment is labeled in mL, how will you know which measurement to use? We can apply this conversion formula: 1 cup = 237mL. This means that ½ cup = 118.5mL. Again, this exact measurement is probably not on the measuring cup. It is probably closest to 125mL, so we will again have to estimate.

Most recipes give guidelines as to how much a single batch will produce. But what if you want more? It seems too time consuming to mix up another batch. What if the recipe makes only one dozen cupcakes and you need three dozen? Clearly, three dozen is three times more than 1 dozen, so we can multiply all the ingredients by three to make a larger batch. It is also important to understand how to multiply fractions. If the cupcake recipe calls for ¾ cup of milk and we want to triple it, we need to know that:

So, we will need 2 and 1/4 cups of milk to make three dozen cupcakes.

This knowledge of fractions is also helpful when we need to make our batch smaller. For example, recipe guidelines approximate that each batch will yield 6 dozen cookies. But, my family is small and I only want 2 dozen cookies. First, we need to see the relationship between 2 and 6. We can see that 2 dozen is one third (1/3) of 6 dozen because 2 x 3 = 6. That means that in order to make only 2 dozen cookies, we will need to use one third of each ingredient. So, if the recipe asks for 2 teaspoons of baking powder, we will only need 2/3 of a teaspoon, since

If we do not have a measuring spoon that is equal to 2/3 teaspoon, we may need to use 1/3 twice, or estimate using ¼.

When recipes indicate how much a particular batch will make, they give a general amount of food. If we are cooking for a group, we need to estimate how much each person will eat and make appropriate amounts of the particular item. For example, if a package of spaghetti makes 1L of cooked spaghetti, will we have enough to feed six people with one package? If not, how much of a second package will we have to use? First, we need to estimate how much each person will eat. We can guess that each person will eat 1 cup of spaghetti, which is 237mL. For convenience sake, we can round this to 250mL. That means that six people will eat 1500mL of spaghetti. If 1L=1000mL, we know that we will need to make one whole package, plus half of the second package to ensure that everyone has enough to eat.

Sometimes, we may not have all the ingredients to make a recipe, but we may have something we can fittingly substitute. How does this affect the measurement amounts in the recipe? For example, let&rsquos imagine we are making Rice Krispie cake. The recipe calls for 32 large marshmallows, but we only have miniature marshmallows. We can still use the small marshmallows, but we will need to estimate how many mini marshmallows would make one large marshmallow, and multiply that number by 32.

What if you want to spice up your chocolate chip cookies by adding almonds and coconut? Your recipe calls for 2 cups of chocolate chips, but you want to add 1/3 cup of almonds and 1/6 cup of coconut. How much chocolate chips do you still have to add? Well, we simply need to subtract, using fractions.

We still need to add 1 and ½ cups of chocolate chips. It is important to remember that when adding and subtracting fractions, we need to use a common denominator.

Weight often affects cooking time. Consider the following hypothetical situation: we are cooking an 8 pound turkey for Christmas dinner. If the turkey needs to thaw in the refrigerator for 24 hours, per 5 pounds, we need to take the turkey out of the freezer in advance. We can use a proportional relation to help us decide how early to thaw the turkey.

The above proportion reads as follows: 5 pounds is to 24 hours as 8 pounds is to x hours. By cross-multiplying and dividing, we can find an answer of 38.4 hours, which is the solution for x.
If we are instructed to cook the turkey for 20 minutes per pound, how long do we need to cook the turkey? Well, 20 minutes per pound for 8 pounds is 20 x 8 = 160 minutes. And, 160 minutes is two hours and 40 minutes. If we only knew the weight of our turkey in kilograms, we would need another conversion formula (kilograms to pounds) to find the weight of the turkey in pounds first, and then apply the recommendations.

We also use math when cooking and baking to estimate the cost of a certain dish. We can understand that cheesecake is more expensive to make than a batch of cookies, particularly when people buy ingredients such as flour, sugar, and butter in bulk and cream cheese is more expensive. When comparing recipes, it may be beneficial to estimate the cost of each recipe.

Mathematical skills are used quite frequently when baking and cooking. It can be very helpful to understand how math affects the quality of culinary in order to make the most delicious meals and treats.

Try this cooking Web Quest that involves math.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.

### Math Error Analysis

I am a math teacher and I have been creating and using math error analysis activities with my students (Grades 5 - 8) for the past 13 years.

Benefits of Math Error Analysis:
Giving students opportunities to identify and correct errors in presented solutions allows them to show their understanding of the mathematical concepts I have just taught.

My Experience:
Every time I incorporate math error analysis activities into my lessons I am always amazed at how quickly it leads to richer discussions and deeper thinking among my students. They suddenly become little math critics and their focus broadens from getting the answer to understanding the process.

Whats Included:
Each of my 'Math Error Analysis Activities' includes 10 real-world WORD PROBLEMS that are solved incorrectly. Students have to IDENTIFY THE ERROR, provide the CORRECT SOLUTION and share a helpful STRATEGY for solving the problem. An ANSWER KEY has also been provided with examples of possible answers.

How to Use:
When I present this activity to my students, I tell them that these are errors that students from my other class made and I need their help to correct them. Students LOVE correcting other students mistakes.

I have my students use these activities with a partner, as a warm-up , as classwork, homework, in math centers and for group work.

I currently have math error analysis activities for more than 30 math topics for grades 5 - 8.

## Separable Differential Equations

If the function $$<>, ight)>>>$$ can be written as the product of the function $$<> ight)>>>$$ (function that depends only on  ) and the function $$ight)>$$ (function that depends only on  ), such a differential equation is called separable.

Let's see how it is solved.

Integrating both sides yields $$intfrac<<>><< ight)>>>=int<> ight)>>>+$$ , where  is an arbitrary constant.

This equation is separable and can be rewrritten as $$frac<<3>><<>^<<4>>>=+<1> ight)>$$ .

Rewriting it a bit, we obtain that $$=-frac<<1>><>]<><<2>><>^<<2>>++>>>>$$ . This is the general solution. To find the particular solution, plug in the initial values and find the constant  .

Now, let's take a look at another example.

There are also 2 particular cases: when $$<> ight)>>>= <1>$$ (the differential equation doesn't contain  ) or $$<> ight)>>>= <1>$$ (the differential equation doesn't contain  ).

Let's do some more practice.

Integrating both sides gives $$<>^<>=+$$ , where  is an arbitrary constant.

And one more final example.

Integrating both sides gives $$=- ight)>>>+$$ , where  is an arbitrary constant.

To find the particular solution, use the initial condition $$ight)>= <5>$$ :