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acm95b_notes - ACM 95b/100b: Ordinary Differential...

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Unformatted text preview: ACM 95b/100b: Ordinary Differential Equations Eric Tai March 14, 2007 Contents 0.1 Linear First Order ODE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 Second Order Linear IVPs 5 1.1 Homogeneous Second Order Linear IVP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Solution Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Solution Methods 6 2.1 Homogeneous Second Order Linear IVPs with Constant Coefficients . . . . . . . . . . . . . . 6 2.2 Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Nonhomogeneous Second Order IVPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Dirac Delta Function 9 4 Heaviside Step Function 9 5 Greens Functions for IVP 9 6 Laplace Transforms 11 6.1 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 6.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.3 Convolution Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.4 Shifting Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 6.5 IVPs with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 6.6 IVPs with Discontinuous Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.7 IVPs with Impulsive Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 6.8 Inverting Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 6.8.1 Evaluating Bromwich Contour Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 16 7 Nonlinear 1st Order ODEs 18 7.1 Existence & Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 7.2 Picards Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 8 Numerical Methods for IVPs 19 8.1 Eulers Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 8.2 Truncation Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 8.3 Error Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 8.4 Removing Implicitness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.5 Explicit Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.6 Runge-Kutta for n-th Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 23 8.7 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1 CONTENTS CONTENTS...
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