In this discussion we will investigate how to solve certain homogeneous systems of linear differential equations. We will also look at a sketch of the solutions.

Example (PageIndex{1})

Consider the system of differential equations

[ x' = x + y onumber ]

[ y' = -2x + 4y. onumber ]

This is a system of differential equations. Clearly the trivial solution ((x = 0) and (y = 0)) is a solution, which is called a *node* for this system. We want to investigate the behavior of the other solutions. Do they approach the origin or are they repelled from it? We can graph the system by plotting direction arrows. We will calculate a few of these arrows and then use a computer to finish the plot.

Consider the point ((0,1)). We have

[x' = 1 ;;; ext{and} ;;; y' = 4 onumber ]

so that

[ dfrac{dy}{dx} = dfrac{4}{1} = 4. onumber ]

We plot a small arrow emanating from ((0,1)) with slope 4.

Consider the point ((1,0)). We have

[x' = 1 ;;; ext{and} ;;; y' = -2 onumber ]

so that

[ dfrac{dy}{dx} = dfrac{-2}{1} = -2. onumber ]

We plot a small arrow emanating from ((1,0)) with slope -2. Below is a computer generated graph. We call the xy-plane the *phase plane* for the differential equation and the plot the *phase portrait*.

Notice that all solutions are repelled from the origin. The origin is called an *unstable equilibrium point*.

Our next task is to find an explicit solution for the system. We write the system as

[ extbf{x}' = A extbf{x} onumber ]

where

[A=egin{pmatrix} 1 &1 -2 & 4 end{pmatrix}. onumber ]

Just as with higher order differential equations, we assume that the solution is in the form

[ extbf{x} = extbf{z}e^{rt} onumber ]

where **x** is a vector solution and **z** is a constant vector. We have

[ extbf{x}' = r extbf{z}e^{rt} onumber ]

so that

[r extbf{z}e^{rt}=A extbf{z}e^{rt}. onumber ]

We can divide by (e^{rt}) to get

[A extbf{z}=r extbf{z}. onumber ]

Finding a solution is equivalent to finding an eigenvalue of the matrix. We can use a calculator to find that the eigenvalues are

[r = 2 ;;; ext{and} ;;; r = 3. onumber ]

To find the constants, we can plug the eigenvalues into

[ A - rI. onumber ]

Plugging in (r = 2) gives

[A-2I = egin{pmatrix} -1 & -1 -2 & 2 end{pmatrix}. onumber ]

The first row tells us that

[ -x + y = 0 onumber ]

and the eigenvector is

[z_2=egin{pmatrix} 1 1 end{pmatrix}. onumber ]

Plugging in (r = 3) gives

[A-3I = egin{pmatrix} -2 & 1 -2 & 1 end{pmatrix}. onumber ]

The first row tells us that

[-2x + y = 0 onumber ]

and the eigenvector is

[ z_3 = egin{pmatrix} 1 2 end{pmatrix}. onumber ]

We can conclude that the general solution is

[egin{pmatrix} x y end{pmatrix} = c_1egin{pmatrix}1 1end{pmatrix}e^{2t} + c_2egin{pmatrix}1 2 end{pmatrix} e^{3t} onumber ]

or that

[ x = c_1e^{2t} + c_2e^{3t} onumber ]

[ y = c_1e^{2t} + 2c_2e^{3t}. onumber ]

There is a direction relationship between the type of node the origin is and the eigenvalues of the matrix. In the example above the eigenvalues are both positive, thus both x and y are repelled from the origin as (t) becomes large. In particular if either of the eigenvalues are positive, then the trajectories repel from the origin. If the eigenvalues had both been negative, then both (x) and (y) approach zero as (t) approaches 0. Hence, all trajectories travel towards the origin for large (t).

The cases where the eigenvalues are complex will be studied in the next discussion.

## 5.2: Homogeneous Systems of Differential Equations - Mathematics

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We learn how to recognize whether or not a first-order equation is exact. We also learn how to solve an exact equation.

### Exact Equations

In this section it’s convenient to write first order differential equations in the form

This equation can be interpreted as where is the independent variable and is the dependent variable, or as where is the independent variable and is the dependent variable. Since the solutions of (eq:2.5.2) and (eq:2.5.3) will often have to be left in implicit form, we’ll say that is an implicit solution of (eq:2.5.1) if every differentiable function that satisfies is a solution of (eq:2.5.2) and every differentiable function that satisfies is a solution of (eq:2.5.3).

Note that a separable equation can be written as (eq:2.5.1) as

We’ll develop a method for solving (eq:2.5.1) under appropriate assumptions on and . This method is an extension of the method of separation of variables. Before stating it we consider an example.

You may think this example is pointless, since concocting a differential equation that has a given implicit solution isn’t particularly interesting. However, it illustrates the next important theorem, which we’ll prove by using implicit differentiation, as in Example example:2.5.1.

Proof Regarding as a function of and differentiating (eq:2.5.6) implicitly with respect to yields On the other hand, regarding as a function of and differentiating (eq:2.5.6) implicitly with respect to yields Thus, (eq:2.5.6) is an implicit solution of (eq:2.5.7) in either of its two possible interpretations.

We’ll say that the equation

is *exact* on an an open rectangle if there’s a function such that and are continuous, and for all in . This usage of “exact” is related to its usage in calculus, where the expression (obtained by substituting (eq:2.5.9) into the left side of (eq:2.5.8)) is the *exact differential of* .

Example example:2.5.1 shows that it’s easy to solve (eq:2.5.8) if it’s exact *and* we know a function that satisfies (eq:2.5.9). The important questions are:

##### Question 1:

Given an equation (eq:2.5.8), how can we determine whether it’s exact?

##### Question 2:

If (eq:2.5.8) is exact, how do we find a function satisfying (eq:2.5.9)?

To discover the answer to Question 1, assume that there’s a function that satisfies (eq:2.5.9) on some open rectangle , and in addition that has continuous mixed partial derivatives and . Then a theorem from calculus implies that

If and , differentiating the first of these equations with respect to and the second with respect to yields From (eq:2.5.10) and (eq:2.5.11), we conclude that a necessary condition for exactness is that . This motivates the next theorem, which we state without proof.

To help you remember the exactness condition, observe that the coefficients of and are differentiated in (eq:2.5.12) with respect to the “opposite” variables that is, the coefficient of is differentiated with respect to , while the coefficient of is differentiated with respect to .

The next example illustrates two possible methods for finding a function that satisfies the condition and if is exact.

*Method 2:* Instead of first integrating (eq:2.5.14) with respect to , we could begin by integrating (eq:2.5.15) with respect to to obtain

where is an arbitrary function of . To determine , we assume that is differentiable and differentiate with respect to , which yields Comparing this with (eq:2.5.14) shows that Integrating this and again taking the constant of integration to be zero yields Substituting this into (eq:2.5.18) yields (eq:2.5.17).

Figure figure:2.5.1 shows a direction field and some integral curves of (eq:2.5.13),

Here’s a summary of the procedure used in Method 1 of this example. You should summarize procedure used in Method 2.

Procedure For Solving An Exact Equation

*Step 1.* Check that the equation

satisfies the exactness condition . If not, don’t go further with this procedure.

*Step 2.* Integrate with respect to to obtain

where is an antiderivative of with respect to , and is an unknown function of .

*Step 3.* Differentiate (eq:2.5.20) with respect to to obtain

*Step 4.* Equate the right side of this equation to and solve for thus,

*Step 5.* Integrate with respect to , taking the constant of integration to be zero, and substitute the result in (eq:2.5.20) to obtain .

*Step 6.* Set to obtain an implicit solution of (eq:2.5.19). If possible, solve for explicitly as a function of .

It’s a common mistake to omit Step 6. However, it’s important to include this step, since isn’t itself a solution of (eq:2.5.19).

Many equations can be conveniently solved by either of the two methods used in Example example:2.5.3. However, sometimes the integration required in one approach is more difficult than in the other. In such cases we choose the approach that requires the easier integration.

Attempting to apply our procedure to an equation that isn’t exact will lead to failure in Step 4, since the function won’t be independent of if , and therefore can’t be the derivative of a function of alone. Here’s an example that illustrates this.

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)

## 5.2: Homogeneous Systems of Differential Equations - Mathematics

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### Introduction to Systems of Differential Equations

Many physical situations are modelled by systems of differential equations in unknown functions, where . The next three examples illustrate physical problems that lead to systems of differential equations. In these examples and throughout this chapter we’ll denote the independent variable by .

A solution with pound of salt per gallon is pumped into from an external source at gal/min, and a solution with pounds of salt per gallon is pumped into from an external source at gal/min. The solution from is pumped into at 2 gal/min, and the solution from is pumped into at gal/min. is drained at gal/min and is drained at 3 gal/min. Let and be the number of pounds of salt in and , respectively, at time . Derive a system of differential equations for and . Assume that both mixtures are well stirred.

receives salt from the external source at the rate of and from at the rate of Therefore

Solution leaves at the rate of 8 gal/min, since 6 gal/min are drained and 2 gal/min are pumped to hence, Eqns. (eq:10.1.1) and (eq:10.1.2) imply that

receives salt from the external source at the rate of and from at the rate of Therefore

Solution leaves at the rate of gal/min, since gal/min are drained and gal/min are pumped to hence, Eqns. (eq:10.1.4) and (eq:10.1.5) imply that

We say that (eq:10.1.3) and (eq:10.1.6) form a *system of two first order equations in two unknowns*, and write them together as

The springs obey Hooke’s law, with spring constants and . Internal friction causes the springs to exert damping forces proportional to the rates of change of their lengths, with damping constants and . Let and be the displacements of the two masses from their equilibrium positions at time , measured positive upward. Derive a system of differential equations for and , assuming that the masses of the springs are negligible and that vertical external forces and also act on the objects.

Let be the Hooke’s law force acting on , and let be the damping force on . Similarly, let and be the Hooke’s law and damping forces acting on . According to Newton’s second law of motion,

When the displacements are and , the change in length of is and the change in length of is . Both springs exert Hooke’s law forces on , while only exerts a Hooke’s law force on . These forces are in directions that tend to restore the springs to their natural lengths. Therefore When the velocities are and , and are changing length at the rates and , respectively. Both springs exert damping forces on , while only exerts a damping force on . Since the force due to damping exerted by a spring is proportional to the rate of change of length of the spring and in a direction that opposes the change, it follows that

and From (eq:10.1.7), Therefore we can rewrite (eq:10.1.11) and (eq:10.1.12) as

According to Newton’s law of gravitation, Earth’s gravitational force on the object is inversely proportional to the square of the distance of the object from Earth’s center, and directed toward the center thus,

where is a constant. To determine , we observe that the magnitude of is Let be Earth’s radius. Since when the object is at Earth’s surface, Therefore we can rewrite (eq:10.1.13) as Now suppose is the only force acting on the object. According to Newton’s second law of motion, that is, Cancelling the common factor and equating components on the two sides of this equation yields the system

#### Rewriting Higher Order Systems as First Order Systems

is called a *first order system*, since the only derivatives occurring in it are first derivatives. The derivative of each of the unknowns may depend upon the independent variable and all the unknowns, but not on the derivatives of other unknowns. When we wish to emphasize the number of unknown functions in (eq:10.1.15) we will say that (eq:10.1.15) is an system.

Systems involving higher order derivatives can often be reformulated as first order systems by introducing additional unknowns. The next two examples illustrate this.

#### Rewriting Scalar Differential Equations as Systems

In this chapter we’ll refer to differential equations involving only one unknown function as *scalar* differential equations. Scalar differential equations can be rewritten as systems of first order equations by the method illustrated in the next two examples.

Since systems of differential equations involving higher derivatives can be rewritten as first order systems by the method used in Examples example:10.1.5 –example:10.1.7 , we’ll consider only first order systems.

#### Numerical Solution of Systems

The numerical methods that we studied in Chapter 3 can be extended to systems, and most differential equation software packages include programs to solve systems of equations. We won’t go into detail on numerical methods for systems however, for illustrative purposes we’ll describe the Runge-Kutta method for the numerical solution of the initial value problem

### Text Source

Trench, William F., ”Elementary Differential Equations” (2013). Faculty Authored and Edited Books & CDs. 8. (CC-BY-NC-SA)

## Linear Homogeneous Systems of Differential Equations with Constant Coefficients – Page 2

We differentiate the first equation and then substitute the derivative (

Express (3

Substituting this into the last equation, we get:

Find the roots of the characteristic equation:

Hence, the general solution of the (2)nd order equation for the variable (

where (

Now we compute the derivative (

To keep integer coefficients, it is convenient to designate: (

### Example 2.

We convert this system to a single (2)nd order equation for the function (xleft( t ight).) Differentiating the first equation and substituting (y’) from the second equation, we have:

Express the variable (y) in terms of (x) and (x’) from the first equation:

Compute the roots of the auxiliary equation:

So, we have one root (lambda = 5) of multiplicity (2.) Consequently, the general solution for the function (xleft( t ight)) is written as

where (

Find the derivative (x’left( t ight)) and substituting it in the first equation of the original system determine the function (yleft( t ight):)

Thus, the general solution is written as

### Example 3.

Differentiating the first equation, we get:

Substitute the derivative (

Express (2

We have obtained a homogeneous equation of the (2)nd order with constant coefficients. As usual, we construct the general solution using the characteristic equation:

As it can be seen, the auxiliary equation has roots in the form of a pair of complex conjugate numbers. The general solution for the function (

where (

Now we find another function (

Substituting (

A. I. Fomin, “Differential Symmetry Algebras for Linear Homogeneous Differential Equations,” Russ. J. Math. Phys. **5**(2), 189–210 (1997).

W.[V.] I. Fushchich and A. G. Nikitin, *Symmetries of Equations of Quantum Mechanics* (Nauka, Moscow, 1990 Allerton Press, Inc., New York, 1994).

A. O. Barut and R. Raczka, *Theory of Group Representations and Applications* (PWN—Polish Scientific Publishers, Warsaw, 1977 Mir, Moscow, 1980).

I. A. Malkin and V. I. Man’ko, *Dynamical Symmetry and Coherent States of Quantum Systems* (Nauka, Moscow, 1990) [in Russian].

D. P. Zhelobenko, *Representations of Reductive Lie Algebras* (Fizmatlit, Moscow, 1994).

D. P. Zhelobenko, “Transvector Algebras in Representation Theory and Dynamic Symmetry,” in: *Group Theoretical Methods in Physics, Vol. II (Yurmala, 1985)* (VNU Sci. Press, Utrecht, 1986) pp. 71–93.

I. S. Krasil’shchik [Krasilshchik], V. V. Lychagin, and A. M. Vinogradov, *Introduction to the Geometry of Nonlinear Differential Equations* (Nauka, Moscow, 1986) [*Geometry of Jet Spaces and Nonlinear Partial Differential Equations*, Gordon & Breach Science Publishers, New York, 1986].

A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khorkova, I. S. Krasil’shchik, A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky [Verbovetskii], and A. M. Vinogradov, *Symmetries and Conservation Laws for Differential Equations of Mathematical Physics*, Transl. Math. Monogr., 182 (Faktorial, Moscow, 1997, American Mathematical Society, Providence, RI, 1999).

L. V. Ovsyannikov, *Group Analysis of Differential Equations* (Nauka, Moscow, 1978 Academic Press, New York a.o., 1982).

N. Kh. Ibragimov, *Transformation Groups in Mathematical Physics* (Nauka, Moscow, 1983) [*Transformation Groups Applied to Mathematical Physics* (D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1985)].

P. J. Olver, *Applications of Lie Groups to Differential Equations* (Mir, Moscow, 1989 Oxford University, Mathematical Institute, Oxford, 1980).

Yu. I. Manin, “Algebraic Aspects of Nonlinear Differential Equations,” Current Problems in Mathematics, Vol. 11, Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat. **11** (Akad. Nauk SSSR Vsesojuz. Inst. Naucn. Tekhn. Informacii, Moscow, 1978), pp. 5–152 [J. Soviet Math. **11** (1), 1–122 (1979)].

S. MacLane, *Homology*, Reprint of the 1st ed. Die Grundlehren der mathematischen Wissenschaften, Band 114 (Springer-Verlag, Berlin-New York, 1967 Mir, Moscow, 1976).

N. N. Bogoliubov [Bogolyubov] and D. V. Shirkov, *Quantum Fields* (Nauka, Moscow, 1993 English transl. of the 1st ed.: Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA, 1983).

A. N. Tikhonov and A. A. Samarskii, *Equations of Mathematical Physics* (4th ed. Nauka, Moscow, 1972 English transl. of the 3rd ed.: Dover Publications, Inc., New York, 1990).

V. S. Vladimirov, *Equations of Mathematical Physics*, Pure and Applied Mathematics, 3 (3rd ed. Nauka, Moscow, 1976 English transl. of the 2nd ed.: Marcel Dekker, Inc., New York, 1971).

L. Hörmander, *An Introduction to Complex Analysis in Several Variables*, 2nd revised ed., North-Holland Mathematical Library, Vol. 7 (North-Holland Publishing Co., Amsterdam-London American Elsevier Publishing Co., Inc., New York, 1973 Russian transl. of the 1st ed.: Mir, Moscow, 1968).

## 5.2: Homogeneous Systems of Differential Equations - Mathematics

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### Problem

### For each of the following,draw suitable compartme…

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In the meantime, our AI Tutor recommends this similar expert step-by-step video covering the same topics.

### Problem 1 Easy Difficulty

In Section 5.2, a model for an epidemic was developed, which led to the system of differential equations in the form

$

frac

$

Use parameter values $eta=0.002$ and $gamma=0.4$, and assume that initially there is only one infective but there are 500 susceptibles. Use MATLAB or Maple to generate the time-dependent plot on the interval $t=[0,20]$.

(a) How many susceptibles never get infected, and what is the maximum number of infectives at any time?

(b) What happens, as time progresses, if the initial number of susceptibles is doubled, $S(0)=$ $1,000 ?$ How many people were infected in total?

(c) Return the initial number of susceptibles to 500. Suppose the transmission coefficient $eta$ is doubled. How does this affect the maximum number of infected individuals? Is this what you expect?

(d) Draw the compartment diagram for the SIR model with an additional dashed line that indicates which rates are also influenced by any other compartments.

Department of Computer Engineering, Baskent University, 06790, Turkey

*** Corresponding author: Nizami A. Gasilov, [email protected]**

**Received** February 2020 **Revised** April 2020 **Published** May 2021 **Early access** June 2020

In this study, we consider a system of homogeneous linear differential equations, the coefficients and initial values of which are constant intervals. We apply the approach that treats an interval problem as a set of real (classical) problems. In previous studies, a system of linear differential equations with real coefficients, but with interval forcing terms and interval initial values was investigated. It was shown that the value of the solution at each time instant forms a convex polygon in the coordinate plane. The motivating question of the present study is to investigate whether the same statement remains true, when the coefficients are intervals. Numerical experiments show that the answer is negative. Namely, at a fixed time, the region formed by the solution's value is not necessarily a polygon. Moreover, this region can be non-convex.

The solution, defined in this study, is compared with the Hukuhara- differentiable solution, and its advantages are exhibited. First, under the proposed concept, the solution always exists and is unique. Second, this solution concept does not require a set-valued, or interval-valued derivative. Third, the concept is successful because it seeks a solution from a wider class of set-valued functions.

##### References:

Ş. E. Amrahov, A. Khastan, N. Gasilov and A. G. Fatullayev, Relationship between Bede-Gal differentiable set-valued functions and their associated support functions, *Fuzzy Sets and Systems*, **295** (2016), 57-71. doi: 10.1016/j.fss.2015.12.002. Google Scholar

J.-P. Aubin and H. Frankowska, *Set-Valued Analysis*, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. Google Scholar

Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores and M. D. Jiménez-Gamero, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, *Fuzzy Sets and Systems*, **219** (2013), 49-67. doi: 10.1016/j.fss.2012.12.004. Google Scholar

T. F. Filippova, Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty, *IFAC PapersOnLine*, **51** (2018), 770-775, http:dx.doi.org/10.1016/j.ifacol.2018.11.452. Google Scholar

N. A. Gasilov and Ş. E. Amrahov, On differential equations with interval coefficients, *Mathematical Methods in the Applied Sciences*, **43** (2020), 1825-1837. doi: 10.1002/mma.6006. Google Scholar

N. A. Gasilov and Ş. E. Amrahov, Solving a nonhomogeneous linear system of interval differential equations, *Soft Computing*, **22** (2018), 3817-3828. doi: 10.1007/s00500-017-2818-x. Google Scholar

N. A. Gasilov and M. Kaya, A method for the numerical solution of a boundary value problem for a linear differential equation with interval parameters, *International Journal of Computational Methods*, **16** (2019), 1850115, 17 pp. doi: 10.1142/S0219876218501153. Google Scholar

E. Hüllermeier, An approach to modeling and simulation of uncertain dynamical systems, *International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems*, **5** (1997), 117-137. doi: 10.1142/S0218488597000117. Google Scholar

R. B. Kearfott and V. Kreinovich, Applications of interval computations: An introduction, *Applications of Interval Computations, Appl. Optim., Kluwer Acad. Publ., Dordrecht*, **3** (1996), 1-22. doi: 10.1007/978-1-4613-3440-8_1. Google Scholar

V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, *Theory of Set Differential Equations in Metric Spaces*, Cambridge Scientific Publishers, Cambridge, 2006. Google Scholar

M. T. Malinowski, Interval differential equations with a second type Hukuhara derivative, *Applied Mathematics Letters*, **24** (2011), 2118-2123. doi: 10.1016/j.aml.2011.06.011. Google Scholar

R. E. Moore, R. B. Kearfott and M. J. Cloud, *Introduction to Interval Analysis*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716. Google Scholar

A. V. Plotnikov and T. A. Komleva, On some properties of bundles of trajectories of a controlled bilinear inclusion, *Ukrainian Mathematical Journal*, **56** (2004), 586-600. doi: 10.1007/s11253-005-0114-x. Google Scholar

A. V. Plotnikov and N. V. Skripnik, Conditions for the existence of local solutions of set-valued differential equations with generalized derivative, *Ukrainian Mathematical Journal*, **65** (2014), 1498-1513. doi: 10.1007/s11253-014-0875-1. Google Scholar

L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, *Nonlinear Anal.*, **71** (2009), 1311-1328. doi: 10.1016/j.na.2008.12.005. Google Scholar

##### References:

Ş. E. Amrahov, A. Khastan, N. Gasilov and A. G. Fatullayev, Relationship between Bede-Gal differentiable set-valued functions and their associated support functions, *Fuzzy Sets and Systems*, **295** (2016), 57-71. doi: 10.1016/j.fss.2015.12.002. Google Scholar

J.-P. Aubin and H. Frankowska, *Set-Valued Analysis*, Systems & Control: Foundations & Applications, 2. Birkhäuser Boston, Inc., Boston, MA, 1990. Google Scholar

Y. Chalco-Cano, A. Rufián-Lizana, H. Román-Flores and M. D. Jiménez-Gamero, Calculus for interval-valued functions using generalized Hukuhara derivative and applications, *Fuzzy Sets and Systems*, **219** (2013), 49-67. doi: 10.1016/j.fss.2012.12.004. Google Scholar

T. F. Filippova, Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty, *IFAC PapersOnLine*, **51** (2018), 770-775, http:dx.doi.org/10.1016/j.ifacol.2018.11.452. Google Scholar

N. A. Gasilov and Ş. E. Amrahov, On differential equations with interval coefficients, *Mathematical Methods in the Applied Sciences*, **43** (2020), 1825-1837. doi: 10.1002/mma.6006. Google Scholar

N. A. Gasilov and Ş. E. Amrahov, Solving a nonhomogeneous linear system of interval differential equations, *Soft Computing*, **22** (2018), 3817-3828. doi: 10.1007/s00500-017-2818-x. Google Scholar

N. A. Gasilov and M. Kaya, A method for the numerical solution of a boundary value problem for a linear differential equation with interval parameters, *International Journal of Computational Methods*, **16** (2019), 1850115, 17 pp. doi: 10.1142/S0219876218501153. Google Scholar

E. Hüllermeier, An approach to modeling and simulation of uncertain dynamical systems, *International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems*, **5** (1997), 117-137. doi: 10.1142/S0218488597000117. Google Scholar

R. B. Kearfott and V. Kreinovich, Applications of interval computations: An introduction, *Applications of Interval Computations, Appl. Optim., Kluwer Acad. Publ., Dordrecht*, **3** (1996), 1-22. doi: 10.1007/978-1-4613-3440-8_1. Google Scholar

V. Lakshmikantham, T. G. Bhaskar and J. V. Devi, *Theory of Set Differential Equations in Metric Spaces*, Cambridge Scientific Publishers, Cambridge, 2006. Google Scholar

M. T. Malinowski, Interval differential equations with a second type Hukuhara derivative, *Applied Mathematics Letters*, **24** (2011), 2118-2123. doi: 10.1016/j.aml.2011.06.011. Google Scholar

R. E. Moore, R. B. Kearfott and M. J. Cloud, *Introduction to Interval Analysis*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898717716. Google Scholar

A. V. Plotnikov and T. A. Komleva, On some properties of bundles of trajectories of a controlled bilinear inclusion, *Ukrainian Mathematical Journal*, **56** (2004), 586-600. doi: 10.1007/s11253-005-0114-x. Google Scholar

A. V. Plotnikov and N. V. Skripnik, Conditions for the existence of local solutions of set-valued differential equations with generalized derivative, *Ukrainian Mathematical Journal*, **65** (2014), 1498-1513. doi: 10.1007/s11253-014-0875-1. Google Scholar

L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, *Nonlinear Anal.*, **71** (2009), 1311-1328. doi: 10.1016/j.na.2008.12.005. Google Scholar

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## MA1513 Chapter 4 System OF Differential Equations Outline

IVP: Substitute initial condition 풚(푡 0 )= 풗,푡 = 푡 0 into general solution.

IVP: Solve for specific values of constant c via GE

IVP: Replace c 1 , c 2 . cn with the specific values into the general solution to obtain particular solution (unique)

#### CONVERTING 2ND ORDER ODE TO 1ST ORDER SDE

Consider 2nd Order ODE (Ordinary Differential Equation) 2y’’ – 5y’ + y = 0 Let y 1 = y(t) and y 2 = y’(t)

Differentiate y 1 and y 2 y 1 ’ = y’(t) = y 2 y 2 ’ = y’’(t) = (manipulate original ODE to make y’’ the subject)

#### 푦 1 ′= 푦 2

5 2 푦 2 Solve as Homogeneous SDE

### EQUILIBRIUM POINTS ON A PHASE PLANE

#### STEPS TO SOLVING DIFFERENT EQUILIBRIUM POINTS

*REAL AND DISTINCT EIGENVALUES

- Find Eigenvalues and Eigenvectors
- Find fundamental set of solutions 풙 1 (푡),풙 2 (푡)
- Write general solution 풚 = 푐 1 풙 1 (푡)+ 푐 2 풙 2 (푡)
- IVP: Find particular solution
- Determine Equilibrium Point (Type and Stability using Eigenvalues)
- Sketch Trajectories

*“Asymptotes”: Use eigenvectors *Parallel to? eigenvector: Substitute values to vert to determine which set of solution is bigger. Bigger value, more parallel.

*REAL AND DISTINCT EIGENVALUES

- Find Eigenvalues and Eigenvectors
- Find fundamental set of solutions 풙 1 (푡),풙 2 (푡)
- Write general solution 풚 = 푐 1 풙 1 (푡)+ 푐 2 풙 2 (푡)
- IVP: Find particular solution
- Determine Equilibrium Point (Type and Stability using Eigenvalues)
- Sketch Trajectories

*“Asymptotes”: Use eigenvectors *Parallel to? eigenvector: Substitute t to vert to check which vert is more significant. Check for positively big t and negatively big t.

## Differential Equations: A Modeling Approach

As an undergraduate text, this textbook offers a very accessible introduction to ODEs through modeling, using a real-world applied motivation for introducing each major topic in this subject. This becomes a natural and easy way to understand the basics of differential equations through growth and decay scenarios, easily comprehensible oscillations, and more.

In no way does this approach cause the author to cut corners. Instead, greater pains are taken to make sure key ideas can be understood from a direct modeling approach. In between these episodes of modeling explanation, required techniques and theory are covered.

The book extends the expected overview, examples and exercises approach with section-embedded “instant exercises”. These are answered exercises with the answers later in the section instead of the back of the book. Occasional but detailed real world case studies, including an interesting one tracking an art forgery, further help to illuminate concepts. Starting from an introduction to ODEs, the text continues on to include chapters on nonhomogenous linear equations, autonomous equations, systems of differential equations, Laplace transforms, and PDEs.