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ans - Contents CHAPTER 1 Introduction EXERCISES 1.1...

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Contents CHAPTER 1: Introduction 1 EXERCISES 1.1: Background, page 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 1 EXERCISES 1.2: Solutions and Initial Value Problems, page 14 . . . . . . . . . . . 3 EXERCISES 1.3: Direction Fields, page 22 . . . . . . . . . . . . . . . . . . . . . . . 10 EXERCISES 1.4: The Approximation Method of Euler, page 28 . . . . . . . . . . . 17 CHAPTER 2: First Order Differential Equations 27 EXERCISES 2.2: Separable Equations, page 46 . . . . . . . . . . . . . . . . . . . . 27 EXERCISES 2.3: Linear Equations, page 54 . . . . . . . . . . . . . . . . . . . . . . 41 EXERCISES 2.4: Exact Equations, page 65 . . . . . . . . . . . . . . . . . . . . . . 59 EXERCISES 2.5: Special Integrating Factors, page 71 . . . . . . . . . . . . . . . . . 72 EXERCISES 2.6: Substitutions and Transformations, page 78 . . . . . . . . . . . . 79 REVIEW PROBLEMS: page 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 CHAPTER 3: Mathematical Models and Numerical Methods Involving First Order Equations 103 EXERCISES 3.2: Compartmental Analysis, page 98 . . . . . . . . . . . . . . . . . . 103 EXERCISES 3.3: Heating and Cooling of Buildings, page 107 . . . . . . . . . . . . 116 EXERCISES 3.4: Newtonian Mechanics, page 115 . . . . . . . . . . . . . . . . . . . 123 EXERCISES 3.5: Electrical Circuits, page 122 . . . . . . . . . . . . . . . . . . . . . 137 EXERCISES 3.6: Improved Euler’s Method, page 132 . . . . . . . . . . . . . . . . . 139 EXERCISES 3.7: Higher Order Numerical Methods: Taylor and Runge-Kutta, page 142 153 CHAPTER 4: Linear Second Order Equations 167 EXERCISES 4.1: Introduction: The Mass-Spring Oscillator, page 159 . . . . . . . . 167 EXERCISES 4.2: Homogeneous Linear Equations; The General Solution, page 167 . 169 EXERCISES 4.3: Auxiliary Equations with Complex Roots, page 177 . . . . . . . . 177 iii

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EXERCISES 4.4: Nonhomogeneous Equations: The Method of Undetermined Coeﬃcients, page 186 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 EXERCISES 4.5: The Superposition Principle and Undetermined Coeﬃcients Revisited, page 192 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 EXERCISES 4.6: Variation of Parameters, page 197 . . . . . . . . . . . . . . . . . . 211 EXERCISES 4.7: Qualitative Considerations for Variable-Coeﬃcient and Nonlinear Equations, page 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 EXERCISES 4.8: A Closer Look at Free Mechanical Vibrations, page 219 . . . . . . 232 EXERCISES 4.9: A Closer Look at Forced Mechanical Vibrations, page 227 . . . . 241 REVIEW PROBLEMS: page 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 CHAPTER 5: Introduction to Systems and Phase Plane Analysis 259 EXERCISES 5.2: Elimination Method for Systems, page 250 . . . . . . . . . . . . . 259 EXERCISES 5.3: Solving Systems and Higher–Order Equations Numerically, page 261 282 EXERCISES 5.4: Introduction to the Phase Plane, page 274 . . . . . . . . . . . . . 293 EXERCISES 5.5: Coupled Mass-Spring Systems, page 284 . . . . . . . . . . . . . . 307 EXERCISES 5.6: Electrical Circuits, page 291 . . . . . . . . . . . . . . . . . . . . . 317 EXERCISES 5.7: Dynamical Systems, Poincar` e Maps, and Chaos, page 301 . . . . . 325 REVIEW PROBLEMS: page 304 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 CHAPTER 6: Theory of Higher Order Linear Differential Equations 341 EXERCISES 6.1: Basic Theory of Linear Differential Equations, page 324 . . . . . . 341 EXERCISES 6.2: Homogeneous Linear Equations with Constant Coeﬃcients, page 331 351 EXERCISES 6.3: Undetermined Coeﬃcients and the Annihilator Method, page 337 361 EXERCISES 6.4: Method of Variation of Parameters, page 341 . . . . . . . . . . . . 375 REVIEW PROBLEMS: page 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 CHAPTER 7: Laplace Transforms 389 EXERCISES 7.2: Definition of the Laplace Transform, page 359 . . . . . . . . . . . 389 EXERCISES 7.3: Properties of the Laplace Transform, page 365 . . . . . . . . . . . 396 EXERCISES 7.4: Inverse Laplace Transform, page 374 . . . . . . . . . . . . . . . . 402 EXERCISES 7.5: Solving Initial Value Problems, page 383 . . . . . . . . . . . . . . 413 EXERCISES 7.6: Transforms of Discontinuous and Periodic Functions, page 395 . . 428 EXERCISES 7.7: Convolution, page 405 . . . . . . . . . . . . . . . . . . . . . . . . 450 EXERCISES 7.8: Impulses and the Dirac Delta Function, page 412 . . . . . . . . . 459 EXERCISES 7.9: Solving Linear Systems with Laplace Transforms, page 416 . . . . 466 REVIEW PROBLEMS: page 418 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 iv
CHAPTER 8: Series Solutions of Differential Equations 491 EXERCISES 8.1:

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