Solving Ordinary Differential Equations

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The study of linear systems of equations given in Chapter ?? provides one motivation for the study of matrices and linear algebra. Linear constant coefficient systems of ordinary differential equations provide a second motivation for this study. In this chapter we show how the phase space geometry of systems of differential equations motivates the idea of eigendirections (or invariant directions) and eigenvalues (or growth rates).

We begin this chapter with a discussion of the theory and application of the simplest of linear differential equations, the linear growth equation, $\dot {x}=\lambda x$ . In Section ??, we solve the linear growth equation and discuss the fact that solutions to differential equations are functions; and we emphasize this point by using MATLAB to graph solutions of $x$ as a function of $t$ . We also illustrate the applicability of this very simple equation with a discussion of compound interest and a simple population model.

In Section ?? we discuss two different ways to plot solutions of differential equations: time series and phase space plots. The first method just plots the graph of a solution as a function of time $t$ , as discussed in Section ??, while the second method is based on thinking of the differential equation as describing how a point moves in space. Both methods are important: time series are typical ways of representing results of experiments and phase space plots are central to a geometric understanding of solutions to differential equations. In Sections ?? and ?? we introduce two MATLAB programs dfield5 (written by John Polking) and pline that illustrate the two methods of plotting the output of a differential equation.

In the optional Section ?? we present one method for solving differential equations analytically where $f(t,x)$ , the right hand side in the ODE, is a product of a function of $x$ and a function of $t$ . This method is called separation of variables and is based on integration theory from calculus. We will see that even these simple differential equations may lead to solutions that are defined only implicitly and not in closed form.

The next two sections introduce planar constant coefficient linear differential equations. In these sections we use the program pplane5 (also written by John Polking) that solves numerically planar systems of differential equations. In Section ?? we discuss uncoupled systems — two independent one dimensional systems like those presented in Section ?? — whose solution geometry in the plane is somewhat more complicated than might be expected. In Section ?? we discuss coupled linear systems. Here we illustrate the existence and nonexistence of eigendirections.

In Section ?? we show how the initial value problems can be solved by building the solution — through the use of superposition as discussed in Section ?? — from simpler solutions. These simpler solutions are ones generated from real eigenvalues and eigenvectors — when they exist. In Section ?? we develop the theory of eigenvalues and characteristic polynomials of $2\times 2$ matrices. (The corresponding theory for $n\times n$ matrices is developed in Chapter ??.)

The method for solving planar constant coefficient linear differential equations with real eigenvalues is summarized in Section ??. This method is based on the material of Sections ?? and ??. The complete discussion of the solutions of linear planar systems of differential equations is given in Chapter ??.

The chapter ends with an optional discussion of Markov chains in Section ??. Markov chains give a method for analyzing branch processes where at each time unit several outcomes are possible, each with a given probability.