Preface |
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vii | |
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1 | (23) |
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1.1 Some Basic Mathematical Models; Direction Fields |
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1 | (8) |
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1.2 Solutions of Some Differential Equations |
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9 | (7) |
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1.3 Classification of Differential Equations |
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16 | (8) |
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2 First-Order Differential Equations |
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24 | (79) |
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2.1 Linear Differential Equations; Method of Integrating Factors |
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24 | (9) |
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2.2 Separable Differential Equations |
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33 | (6) |
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2.3 Modeling with First-Order Differential Equations |
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39 | (12) |
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2.4 Differences Between Linear and Nonlinear Differential Equations |
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51 | (7) |
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2.5 Autonomous Differential Equations and Population Dynamics |
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58 | (12) |
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2.6 Exact Differential Equations and Integrating Factors |
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70 | (6) |
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2.7 Numerical Approximations: Euler's Method |
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76 | (7) |
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2.8 The Existence and Uniqueness Theorem |
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83 | (8) |
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2.9 First-Order Difference Equations |
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91 | (12) |
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3 Second-Order Linear Differential Equations |
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103 | (66) |
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3.1 Homogeneous Differential Equations with Constant Coefficients |
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103 | (7) |
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3.2 Solutions of Linear Homogeneous Equations; the Wronskian |
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110 | (10) |
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3.3 Complex Roots of the Characteristic Equation |
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120 | (7) |
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3.4 Repeated Roots; Reduction of Order |
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127 | (6) |
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3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients |
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133 | (9) |
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3.6 Variation of Parameters |
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142 | (5) |
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3.7 Mechanical and Electrical Vibrations |
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147 | (12) |
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3.8 Forced Periodic Vibrations |
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159 | (10) |
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4 Higher-Order Linear Differential Equations |
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169 | (20) |
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4.1 General Theory of nth Order Linear Differential Equations |
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169 | (5) |
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4.2 Homogeneous Differential Equations with Constant Coefficients |
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174 | (7) |
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4.4 The Method of Undetermined Coefficients |
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181 | (4) |
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4.5 The Method of Variation of Parameters |
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185 | (4) |
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5 Series Solutions of Second-Order Linear Equations |
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189 | (52) |
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5.1 Review of Power Series |
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189 | (6) |
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5.2 Series Solutions Near an Ordinary Point, Part I |
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195 | (10) |
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5.3 Series Solutions Near an Ordinary Point, Part II |
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205 | (6) |
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5.4 Euler Equations; Regular Singular Points |
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211 | (8) |
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5.5 Series Solutions Near a Regular Singular Point, Part I |
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219 | (5) |
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5.6 Series Solutions Near a Regular Singular Point, Part II |
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224 | (6) |
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230 | (11) |
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241 | (40) |
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6.1 Definition of the Laplace Transform |
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241 | (7) |
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6.2 Solution of Initial Value Problems |
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248 | (9) |
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257 | (7) |
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6.4 Differential Equations with Discontinuous Forcing Functions |
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264 | (6) |
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270 | (5) |
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6.6 The Convolution Integral |
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275 | (6) |
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7 Systems of First-Order Linear Equations |
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281 | (73) |
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281 | (5) |
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286 | (9) |
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7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors |
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295 | (9) |
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7.4 Basic Theory of Systems of First-Order Linear Equations |
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304 | (5) |
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7.5 Homogeneous Linear Systems with Constant Coefficients |
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309 | (10) |
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7.6 Complex-Valued Eigenvalues |
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319 | (10) |
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329 | (8) |
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337 | (8) |
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7.9 Nonhomogeneous Linear Systems |
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345 | (9) |
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354 | (34) |
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8.1 The Euler or Tangent Line Method |
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354 | (9) |
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8.2 Improvements on the Euler Method |
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363 | (4) |
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8.3 The Runge-Kutta Method |
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367 | (4) |
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371 | (5) |
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8.5 Systems of First-Order Equations |
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376 | (2) |
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8.6 More on Errors; Stability |
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378 | (10) |
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9 Nonlinear Differential Equations and Stability |
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388 | (75) |
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9.1 The Phase Plane: Linear Systems |
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388 | (10) |
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9.2 Autonomous Systems and Stability |
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398 | (9) |
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9.3 Locally Linear Systems |
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407 | (10) |
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417 | (11) |
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9.5 Predator-Prey Equations |
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428 | (7) |
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9.6 Liapunov's Second Method |
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435 | (9) |
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9.7 Periodic Solutions and Limit Cycles |
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444 | (10) |
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9.8 Chaos and Strange Attractors: The Lorenz Equations |
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454 | (9) |
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10 Partial Differential Equations and Fourier Series |
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463 | (66) |
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10.1 Two-Point Boundary Value Problems |
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463 | (6) |
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469 | (8) |
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10.3 The Fourier Convergence Theorem |
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477 | (5) |
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10.4 Even and Odd Functions |
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482 | (6) |
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10.5 Separation of Variables; Heat Conduction in a Rod |
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488 | (8) |
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10.6 Other Heat Conduction Problems |
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496 | (8) |
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10.7 The Wave Equation: Vibrations of an Elastic String |
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504 | (10) |
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514 | (15) |
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11 Boundary Value Problems and Sturm-Liouville Theory |
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529 | (44) |
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11.1 The Occurrence of Two-Point Boundary Value Problems |
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529 | (6) |
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11.2 Sturm-Liouville Boundary Value Problems |
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535 | (10) |
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11.3 Nonhomogeneous Boundary Value Problems |
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545 | (11) |
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11.4 Singular Sturm-Liouville Problems |
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556 | (6) |
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11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion |
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562 | (4) |
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11.6 Series of Orthogonal Functions: Mean Convergence |
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566 | (7) |
Answers to Problems |
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573 | (35) |
Index |
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608 | |