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Chapter 5 : Systems of Differential Equations

To this point we’ve only looked at solving single differential equations. However, many “real life” situations are governed by a system of differential equations. Consider the population problems that we looked at back in the modeling section of the first order differential equations chapter. In these problems we looked only at a population of one species, yet the problem also contained some information about predators of the species. We assumed that any predation would be constant in these cases. However, in most cases the level of predation would also be dependent upon the population of the predator. So, to be more realistic we should also have a second differential equation that would give the population of the predators. Also note that the population of the predator would be, in some way, dependent upon the population of the prey as well. In other words, we would need to know something about one population to find the other population. So to find the population of either the prey or the predator we would need to solve a system of at least two differential equations.

The next topic of discussion is then how to solve systems of differential equations. However, before doing this we will first need to do a quick review of Linear Algebra. Much of what we will be doing in this chapter will be dependent upon topics from linear algebra. This review is not intended to completely teach you the subject of linear algebra, as that is a topic for a complete class. The quick review is intended to get you familiar enough with some of the basic topics that you will be able to do the work required once we get around to solving systems of differential equations.

Here is a brief listing of the topics covered in this chapter.

Review : Systems of Equations – In this section we will give a review of the traditional starting point for a linear algebra class. We will use linear algebra techniques to solve a system of equations as well as give a couple of useful facts about the number of solutions that a system of equations can have.

Review : Matrices and Vectors – In this section we will give a brief review of matrices and vectors. We will look at arithmetic involving matrices and vectors, finding the inverse of a matrix, computing the determinant of a matrix, linearly dependent/independent vectors and converting systems of equations into matrix form.

Review : Eigenvalues and Eigenvectors – In this section we will introduce the concept of eigenvalues and eigenvectors of a matrix. We define the characteristic polynomial and show how it can be used to find the eigenvalues for a matrix. Once we have the eigenvalues for a matrix we also show how to find the corresponding eigenvalues for the matrix.

Systems of Differential Equations – In this section we will look at some of the basics of systems of differential equations. We show how to convert a system of differential equations into matrix form. In addition, we show how to convert an \(n^{ \text{th}}\) order differential equation into a system of differential equations.

Solutions to Systems – In this section we will a quick overview on how we solve systems of differential equations that are in matrix form. We also define the Wronskian for systems of differential equations and show how it can be used to determine if we have a general solution to the system of differential equations.

Phase Plane – In this section we will give a brief introduction to the phase plane and phase portraits. We define the equilibrium solution/point for a homogeneous system of differential equations and how phase portraits can be used to determine the stability of the equilibrium solution. We also show the formal method of how phase portraits are constructed.

Real Eigenvalues – In this section we will solve systems of two linear differential equations in which the eigenvalues are distinct real numbers. We will also show how to sketch phase portraits associated with real distinct eigenvalues (saddle points and nodes).

Complex Eigenvalues – In this section we will solve systems of two linear differential equations in which the eigenvalues are complex numbers. This will include illustrating how to get a solution that does not involve complex numbers that we usually are after in these cases. We will also show how to sketch phase portraits associated with complex eigenvalues (centers and spirals).

Repeated Eigenvalues – In this section we will solve systems of two linear differential equations in which the eigenvalues are real repeated (double in this case) numbers. This will include deriving a second linearly independent solution that we will need to form the general solution to the system. We will also show how to sketch phase portraits associated with real repeated eigenvalues (improper nodes).

Nonhomogeneous Systems – In this section we will work quick examples illustrating the use of undetermined coefficients and variation of parameters to solve nonhomogeneous systems of differential equations. The method of undetermined coefficients will work pretty much as it does for nth order differential equations, while variation of parameters will need some extra derivation work to get a formula/process we can use on systems.

Laplace Transforms – In this section we will work a quick example illustrating how Laplace transforms can be used to solve a system of two linear differential equations.

Modeling – In this section we’ll take a quick look at some extensions of some of the modeling we did in previous chapters that lead to systems of differential equations. In particular we will look at mixing problems in which we have two interconnected tanks of water, a predator-prey problem in which populations of both are taken into account and a mechanical vibration problem with two masses, connected with a spring and each connected to a wall with a spring.