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Unlike most texts in differential equations, this textbook gives an early presentation of the Laplace transform, which is then used to motivate and develop many of the remaining differential equation concepts for which it is particularly well suited. For example, the standard solution methods for constant coefficient linear differential equations are immediate and simplified, and solution methods for constant coefficient systems are streamlined. By introducing the Laplace transform early in the text, students become proficient in its use while at the same time learning the standard topics in differential equations. The text also includes proofs of several important theorems that are not usually given in introductory texts. These include a proof of the injectivity of the Laplace transform and a proof of the existence and uniqueness theorem for linear constant coefficient differential equations. Along with its unique traits, this text contains all the topics needed for a standard three- or four-hour, sophomore-level differential equations course for students majoring in science or engineering. These topics include: first order differential equations, general linear differential equations with constant coefficients, second order linear differential equations with variable coefficients, power series methods, and linear systems of differential equations. It is assumed that the reader has had the equivalent of a one-year course in college calculus.

William A. Adkins and Mark G. Davidson are currently professors of mathematics at Louisiana State University.

First Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1 An Introduction to Differential Equations . . . . . . . . . . . . . . . . . . 7
1.2 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3 Separable Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.4 Linear First Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.5 Substitutions; Homogeneous and Bernoulli Equations . . . . . . . . 59
1.6 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
1.7 Existence and Uniqueness Theorems . . . . . . . . . . . . . . . . . . . . . . . 71
2 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.1 Definitions, Basic Formulas, and Principles . . . . . . . . . . . . . . . . . 90
2.2 Partial Fractions: A Recursive Method for Linear Terms . . . . . . 107
2.3 Partial Fractions: A Recursive Method for Irreducible
Quadratics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.4 Laplace Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.5 Exponential Polynomials and Laplace Transform
Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
2.6 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
2.7 Laplace Inversion involving Irreducible Quadratics** . . . . . . . . . 157
2.8 Table of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
2.9 Table of Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
3 Second Order Constant Coefficient Linear Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.1 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
3.2 Consequences of Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
3.3 Linear Homogeneous Differential Equations . . . . . . . . . . . . . . . . . 184
3.4 The Method of Undetermined Coefficients . . . . . . . . . . . . . . . . . . 188
3.5 The Incomplete Partial Fraction Method . . . . . . . . . . . . . . . . . . . 195
3.6 Spring Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4 Contents
4 Second Order Linear Differential Equations . . . . . . . . . . . . . . . . 213
4.1 The Existence and Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . 214
4.2 The Homogeneous Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
4.3 The Cauchy-Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
4.4 Laplace Transform Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.5 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
4.6 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
5 Laplace Transform II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
5.1 Calculus of Discontinuous Functions . . . . . . . . . . . . . . . . . . . . . . . 252
5.2 The Heaviside class ℋ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
5.3 Laplace Transform Method for f(t) ∈ ℋ . . . . . . . . . . . . . . . . . . . . 277
5.4 The Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
5.5 Impulse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
5.6 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
5.7 Undamped Motion with Periodic Input . . . . . . . . . . . . . . . . . . . . . 308
5.8 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
6 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.1 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
6.2 Systems of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
6.3 Invertible Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
6.4 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
7 Linear Systems of Differential Equations . . . . . . . . . . . . . . . . . . . 365
7.1 Linear Systems of Differential Equations . . . . . . . . . . . . . . . . . . . . 367
7.2 The Laplace Transform Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
7.3 The Matrix Exponential and its Computation . . . . . . . . . . . . . . . 390
A APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
A.1 The Laplace Transform is Injective . . . . . . . . . . . . . . . . . . . . . . . . 403
A.2 The Linear Independence of ℬq . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405
B Selected Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
C Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
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