In Chapter 4, we discussed several techniques for solving higher-order differential equations. In this chapter, we illustrate how some of these methods can be used to solve initial-value problems that model physical situations.

In Chapter 7, we discuss several applications of systems of differential equations. These include standard linear applications that involve mechanical and electrical systems as well as diffusion and population problems. We also discuss several nonlinear applications including predator–prey interactions, food chains in the chemostat, and curvature, which is a topic from differential geometry.

The purpose of Differential Equations with Mathematica, fifth edition, is twofold. First, we introduce and discuss the topics covered in typical undergraduate and beginning graduate courses in ordinary and partial differential equations including topics such as Laplace transforms, Fourier series, eigenvalue problems, and boundary-value problems. Second, we illustrate how Mathematica is used to enhance the study of differential equations not only by eliminating the computational difficulties but also by overcoming the visual limitations associated with the explicit solutions to differential equations. In each chapter, we first present the material in a manner similar to most differential equations texts and then illustrate how Mathematica can be used to solve some typical problems. For example, in Chapter 2, we introduce the topic of first-order equations. First we show how to solve certain types of problems by hand, and then show how Mathematica can be used to assist in the same solution procedures. Finally, we illustrate how Mathematica commands like DSolve and NDSolve can be used to solve some frequently encountered equations exactly and/or numerically. In Chapter 3, we discuss some applications of first-order equations. Since we are experienced and understand the methods of solution covered in Chapter 2, we make use of DSolve and similar commands to obtain solutions. In doing so we are able to emphasize the applications themselves as opposed to becoming bogged down in calculations.

We will devote a considerable amount of time in this text to developing explicit, implicit, numerical, and graphical solutions of differential equations. In this chapter we introduce frequently encountered forms of first-order ordinary differential equations and methods to construct explicit, numerical, and graphical solutions of them. Several of the equations along with the methods of solution discussed here will be used in subsequent chapters of the text.

In Chapters 2 and 3 we saw that first-order differential equations can be used to model a variety of physical situations. However, many physical situations need to be modeled by higher-order differential equations. In this chapter, we discuss several methods for solving higher-order linear differential equations.

Chapter 10 introduces the separation of variables technique. The emphasis of the chapter is on solving the one-dimensional heat equation, the one-dimensional wave equation, D'Alembert's solution to the one-dimensional wave equation, Laplace's equation, Laplace's equation in a circular region, and the wave equation in a circular region.

When the space shuttle was launched from the Kennedy Space Center, its escape velocity could be determined by solving a first-order ordinary differential equation. The same can be said for finding the flow of electromagnetic forces, the temperature of a cup of coffee, and the population of a species, as well as numerous other applications. In this chapter, we show how these problems can be expressed as first-order equations. We will focus our attention on setting up the problems and explaining the meaning of the subsequent solutions, because the techniques for solving these problems were discussed in Chapter 2.

Because of their importance in the study of systems of linear equations, we now review matrices and the operations associated with them.

In previous chapters, we have seen that many physical situations can be modeled by either ordinary differential equations or systems of ordinary differential equations. However, to understand the motion of a string at a particular location and at a particular time, the temperature in a thin wire at a particular location and a particular time, or the electrostatic potential at a point on a plate, we must solve partial differential equations as each of these quantities depends on (at least) two independent variables.

In previous chapters we have investigated solving the nth-order linear equation(8.1)an(t)y(n)+an−1(t)y(n−1)+⋯+a2(t)y″+a1(t)y′+a0(t)y=f(t) for y. We have seen that if the coefficients ai(t) of (8.1) are numbers, we can find a general solution of the equation by first solving the characteristic equation of the corresponding homogeneous equation, forming a general solution of the corresponding homogeneous equation, and then finding a particular solution to the nonhomogeneous equation. If the coefficients ai(t) are not constants, we have learned that solving (8.1) may be substantially more difficult than in other cases, such as when the functions ai(t) are constants. For example, when Eq. (8.1) is a Cauchy–Euler equation, techniques used to solve the case when (8.1) has constant coefficients can be used to solve the equation. In other situations, we might be able to use a series to find a solution of the equation. Regardless, in all these situations the forcing function f(t) has been a smooth function. If f(t) is not a smooth function, such as when f(t) is a piecewise defined or a periodic function with discontinuities, solving Eq. (8.1) is substantially more difficult to solve using the techniques that we have discussed. In this chapter, we discuss a technique that transforms Eq. (8.1) or systems of linear differential equations into an algebraic equation or equations that can sometimes be solved so that a solution to the differential equation or system of differential equations can be obtained.