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# DE_Complete - DIFFERENTIAL EQUATIONS Paul Dawkins...

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DIFFERENTIAL EQUATIONS Paul Dawkins

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Differential Equations © 2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx Table of Contents Preface ............................................................................................................................................. 3 Outline ............................................................................................................................................ iv Basic Concepts ................................................................................................................................ 1 Introduction ................................................................................................................................................ 1 Definitions .................................................................................................................................................. 2 Direction Fields .......................................................................................................................................... 8 Final Thoughts ......................................................................................................................................... 19 First Order Differential Equations ............................................................................................. 20 Introduction .............................................................................................................................................. 20 Linear Differential Equations ................................................................................................................... 21 Separable Differential Equations .............................................................................................................. 34 Exact Differential Equations .................................................................................................................... 45 Bernoulli Differential Equations .............................................................................................................. 56 Substitutions ............................................................................................................................................. 63 Intervals of Validity ................................................................................................................................. 72 Modeling with First Order Differential Equations ................................................................................... 77 Equilibrium Solutions .............................................................................................................................. 90 Euler’s Method ......................................................................................................................................... 94 Second Order Differential Equations ....................................................................................... 102 Introduction ............................................................................................................................................ 102 Basic Concepts ....................................................................................................................................... 104 Real, Distinct Roots ............................................................................................................................... 109 Complex Roots ....................................................................................................................................... 113 Repeated Roots ....................................................................................................................................... 118 Reduction of Order ................................................................................................................................. 122 Fundamental Sets of Solutions ............................................................................................................... 126 More on the Wronskian .......................................................................................................................... 131 Nonhomogeneous Differential Equations .............................................................................................. 137 Undetermined Coefficients .................................................................................................................... 139 Variation of Parameters .......................................................................................................................... 156 Mechanical Vibrations ........................................................................................................................... 162 Laplace Transforms ................................................................................................................... 181 Introduction ............................................................................................................................................ 181 The Definition ........................................................................................................................................ 183 Laplace Transforms ................................................................................................................................ 187 Inverse Laplace Transforms ................................................................................................................... 191 Step Functions ........................................................................................................................................ 202 Solving IVP’s with Laplace Transforms ................................................................................................ 215 Nonconstant Coefficient IVP’s .............................................................................................................. 222 IVP’s With Step Functions ..................................................................................................................... 226 Dirac Delta Function .............................................................................................................................. 233 Convolution Integrals ............................................................................................................................. 236 Systems of Differential Equations ............................................................................................ 241 Introduction ............................................................................................................................................ 241 Review : Systems of Equations .............................................................................................................. 243 Review : Matrices and Vectors .............................................................................................................. 249 Review : Eigenvalues and Eigenvectors ................................................................................................. 259 Systems of Differential Equations .......................................................................................................... 269 Solutions to Systems .............................................................................................................................. 273 Phase Plane ............................................................................................................................................. 275 Real, Distinct Eigenvalues ..................................................................................................................... 280 Complex Eigenvalues ............................................................................................................................. 290 Repeated Eigenvalues ............................................................................................................................ 296
Differential Equations © 2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx Nonhomogeneous Systems .................................................................................................................... 303 Laplace Transforms ................................................................................................................................ 307 Modeling ................................................................................................................................................ 309 Series Solutions to Differential Equations ............................................................................... 318 Introduction ............................................................................................................................................ 318 Review : Power Series ............................................................................................................................ 319 Review : Taylor Series ........................................................................................................................... 327 Series Solutions to Differential Equations ............................................................................................. 330 Euler Equations ...................................................................................................................................... 340 Higher Order Differential Equations ....................................................................................... 346 Introduction ............................................................................................................................................ 346 Basic Concepts for n th Order Linear Equations ...................................................................................... 347 Linear Homogeneous Differential Equations ......................................................................................... 350 Undetermined Coefficients .................................................................................................................... 355 Variation of Parameters .......................................................................................................................... 357 Laplace Transforms ................................................................................................................................ 363 Systems of Differential Equations .......................................................................................................... 365 Series Solutions ...................................................................................................................................... 370 Boundary Value Problems & Fourier Series ........................................................................... 374 Introduction ............................................................................................................................................ 374 Boundary Value Problems .................................................................................................................... 375 Eigenvalues and Eigenfunctions ............................................................................................................ 381 Periodic Functions, Even/Odd Functions and Orthogonal Functions ..................................................... 398 Fourier Sine Series ................................................................................................................................. 406 Fourier Cosine Series ............................................................................................................................. 417 Fourier Series ......................................................................................................................................... 426 Convergence of Fourier Series ............................................................................................................... 434 Partial Differential Equations ................................................................................................... 440 Introduction ............................................................................................................................................ 440 The Heat Equation .................................................................................................................................. 442 The Wave Equation ................................................................................................................................ 449 Terminology ........................................................................................................................................... 451 Separation of Variables .......................................................................................................................... 454 Solving the Heat Equation ...................................................................................................................... 465 Heat Equation with Non-Zero Temperature Boundaries ........................................................................ 478 Laplace’s Equation ................................................................................................................................. 481 Vibrating String ...................................................................................................................................... 492 Summary of Separation of Variables ..................................................................................................... 495

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Differential Equations © 2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx Preface Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my “class notes”, they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations.
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