| Preface |
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ix | |
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1 | (6) |
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1 | (2) |
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1.3 More on direction fields: Isoclines |
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3 | (4) |
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Chapter 2 First-Order Equations |
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7 | (32) |
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7 | (3) |
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10 | (2) |
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2.3 Applications: Time of death, time at depth, and ancient timekeeping |
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12 | (6) |
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2.4 Existence and uniqueness theorems |
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18 | (3) |
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2.5 Population and financial models |
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21 | (4) |
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2.6 Qualitative solutions of autonomous equations |
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25 | (4) |
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29 | (5) |
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34 | (5) |
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Chapter 3 Numerical Methods |
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39 | (12) |
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39 | (4) |
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3.2 Improving Euler's method: The Heun and Runge-Kutta Algorithms |
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43 | (2) |
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3.3 Optical illusions and other applications |
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45 | (6) |
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Chapter 4 Higher-Order Linear Homogeneous Equations |
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51 | (24) |
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4.1 Introduction to second-order equations |
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51 | (5) |
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56 | (4) |
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60 | (2) |
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4.4 Constant coefficient second-order equations |
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62 | (2) |
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4.5 Repeated roots and reduction of order |
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64 | (4) |
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4.6 Higher-order equations |
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68 | (1) |
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4.7 Higher-order constant coefficient equations |
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69 | (3) |
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4.8 Modeling with second-order equations |
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72 | (3) |
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Chapter 5 Higher-Order Linear Nonhomogeneous Equations |
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75 | (16) |
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5.1 Introduction to nonhomogeneous equations |
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75 | (3) |
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5.2 Annihilating operators |
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78 | (5) |
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5.3 Applications of nonhomogeneous equations |
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83 | (3) |
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86 | (5) |
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Chapter 6 Laplace Transforms |
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91 | (24) |
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91 | (4) |
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6.2 The inverse Laplace transform |
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95 | (1) |
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6.3 Solving initial value problems with Laplace transforms |
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96 | (5) |
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101 | (3) |
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6.5 Laplace transforms, simple systems, and Iwo Jima |
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104 | (3) |
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107 | (3) |
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110 | (5) |
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Chapter 7 Power Series Solutions |
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115 | (30) |
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7.2 Review of power series |
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115 | (2) |
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117 | (9) |
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7.4 Nonpolynomial coefficients |
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126 | (2) |
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7.5 Regular singular points |
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128 | (11) |
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139 | (6) |
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Chapter 8 Linear Systems I |
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145 | (42) |
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8.1 Nelson at Trafalgar and phase portraits |
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145 | (5) |
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8.2 Vectors, vector fields, and matrices |
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150 | (2) |
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8.3 Eigenvalues and eigenvectors |
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152 | (4) |
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8.4 Solving linear systems |
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156 | (6) |
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8.5 Phase portraits via ray solutions |
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162 | (1) |
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8.6 More on phase portraits: Saddle points and nodes |
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163 | (6) |
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8.7 Complex and repeated eigenvalues |
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169 | (4) |
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8.8 Applications: Compartment models |
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173 | (6) |
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8.9 Classifying equilibrium points |
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179 | (8) |
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Chapter 9 Linear Systems II |
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187 | (22) |
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9.1 The matrix exponential, Part I |
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187 | (8) |
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9.2 A return to the Existence and Uniqueness Theorem |
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195 | (2) |
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9.3 The matrix exponential, Part II |
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197 | (5) |
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9.4 Nonhomogeneous constant coefficient systems |
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202 | (5) |
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9.5 Periodic forcing and the steady-state solution |
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207 | (2) |
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Chapter 10 Nonlinear Systems |
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209 | (36) |
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10.1 Introduction: Darwin's finches |
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209 | (5) |
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10.2 Linear approximation: The major cases |
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214 | (8) |
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10.3 Linear approximation: The borderline cases |
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222 | (3) |
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10.4 More on interacting populations |
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225 | (5) |
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10.5 Modeling the spread of disease |
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230 | (4) |
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10.6 Hamiltonians, gradient systems, and Lyapunov functions |
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234 | (5) |
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239 | (3) |
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10.8 Cycles and limit cycles |
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242 | (3) |
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Chapter 11 Partial Differential Equations and Fourier Series |
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245 | |
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11.2 Boundary value problems |
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245 | (2) |
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11.3 Partial differential equations: A first look |
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247 | (3) |
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11.4 Advection and diffusion |
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250 | (3) |
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11.5 Functions as vectors |
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253 | (2) |
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255 | (8) |
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263 | (4) |
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11.8 The wave equation: Separation of variables |
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267 | (5) |
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11.9 The wave equation: D'Alembert's method |
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272 | (5) |
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277 | |
