Differential Equations With Graphical and Numerical Methods

Preface Some time ago I searched for a textbook for a sophomore course in differential equations that would combine analytical (algebraic) methods of solution with graphical and numerical methods in a unified way. Some texts made computer graphics the center of the course and left out such topics as variation of parameters and infinite series. Other texts retained the traditional topics, but the graphics seemed to be grafted on as an afterthought. This book is an outgrowth of this failed search. The book retains almost all the traditional canon of differential equations, but it employs graphical and numerical methods from the outset, both as methods of solution and as means of illuminating concepts. To employ graphical and numerical methods from the start, it was necessary to make first-order systems and reduction to first-order systems the focal point. First-order systems form the core subject matter of Chapters 1 through 5 and Chapter 7. Chapter 6 covers power series solutions, but even here first-order systems make a brief appearance in order to make clear why points at which the leading coefficient of a linear differential equation vanishes must be considered singular. Through first-order systems, solutions can easily be presented graphically with today's computer resources. This opens the way for visual interpretation of solutions and fields. First-order systems also provide the unified means of applying numerical methods to a very wide range of differential equations. Because of this, differential equations can be investigated that could not be considered in times gone by. Models of competing species, the pendulum, and the tunnel diode oscillator are taken up early in the text. In spite of the emphasis on first-order systems, I have not neglected the basics of analytic solutions. Separable, linear, and exact equations are solved in the study of a single first-order equation in Chapter 2, and higher-order constant coefficient linear equations are treated in Chapter 4. However, the knowledge of first-order systems developed in Chapter 3 is used to establish the strategy for solving higher-order linear equations. Power series methods are also not neglected. Indeed, they cannot be, since they are needed in the solution of partial differential equations, which is the subject of Chapter 9. Chapter 9 presents the solution of partial differential equations through the method of separation of variables and Fourier series. Chapter 10 introduces the reader to numerical methods of solution for partial differential equations. These two chapters were more difficult to write than the others because there is no unifying theme, such as first-order systems for ordinary differential equations. Nonetheless, graphics and numerical methods have been employed to help clarify ideas and to extend the range of equations solved. Computer algebra systems (CAS) such as Mathematica, Maple or MatLab (the Three M's) are used to advantage to illustrate convergence of Fourier series, graph modes of vibration for drumheads, and animate solutions. The chapter on numerical methods for partial differential equations is, I think, new in a book of this type. However, I believe it is entirely in keeping with the theme of this book and the availability of powerful computing resources. The use of a CAS makes the instability of some of the finite difference methods easy to explore, and it makes possible the exploration of some nonlinear partial differential equations. Chapter 8 is a traditional treatment of the Laplace transform. The Laplace transform does not call for graphical or numerical methods, but I thought it important to include the Laplace transform because it is such an elegant way of dealing with constant coefficient linear equations and discontinuous forcing functions. A large proportion of the exercises call for the use of a computer. The necessary software is available at the Prentice Hall web site:[read more]