Math2065Fall09

Math2065Fall09 - William A. Adkins, Mark G. Davidson...

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Unformatted text preview: William A. Adkins, Mark G. Davidson ORDINARY DIFFERENTIAL EQUATIONS August 16, 2009 Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo Contents 1 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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Math2065Fall09 - William A. Adkins, Mark G. Davidson...

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