Elementary Differential Equations and Boundary Value Problems 10th Edition
Some Basic Mathematical Models; Direction Fields
Before embarking on a serious study of differential equations (for example, by reading this book or major portions of it), you should have some idea of the possible benefits to be gained by doing so. For some students the intrinsic interest of the subject itself is enough motivation, but for most it is the likelihood of important applications to other fields that makes the undertaking worthwhile.
Many of the principles, or laws, underlying the behavior of the natural world are statements or relations involving rates at which things happen. When expressed in mathematical terms, the relations are equations and the rates are derivatives. Equations containing derivatives are differential equations. Therefore, to understand and to investigate problems involving the motion of fluids, the flow of current in electric circuits, the dissipation of heat in solid objects, the propagation and detection of seismic waves, or the increase or decrease of populations, among many others, it is necessary to know something about differential equations.
A differential equation that describes some physical process is often called a mathematical model of the process, and many such models are discussed throughout this book. In this section we begin with two models leading to equations that are easy to solve. It is noteworthy that even the simplest differential equations provide useful models of important physical processes.
1 Introduction 1
2 First-Order Differential Equations 24
3 Second-Order Linear Differential Equations 103
4 Higher-Order Linear Differential Equations 169
5 Series Solutions of Second-Order Linear Equations 189
6 The Laplace Transform 241
7 Systems of First-Order Linear Equations 281
8 Numerical Methods 354
9 Nonlinear Differential Equations and Stability 388
10 Partial Differential Equations and Fourier Series 463
11 Boundary Value Problems and Sturm-Liouville Theory 529
ANSWERS TO PROBLEMS 573
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