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1 First-Order Differential Equations |
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1 | (64) |
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1.1 Modeling via Differential Equations |
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2 | (4) |
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1.2 Analytic Technique: Separation of Variables |
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6 | (10) |
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1.3 Qualitative Technique: Slope Fields |
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16 | (4) |
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1.4 Numerical Technique: Euler's Method |
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20 | (4) |
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1.5 Existence and Uniqueness of Solutions |
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24 | (3) |
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1.6 Equilibria and the Phase Line |
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27 | (6) |
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33 | (5) |
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38 | (8) |
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1.9 Integrating Factors for Linear Equations |
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46 | (9) |
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55 | (10) |
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65 | (26) |
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66 | (2) |
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2.2 The Geometry of Systems |
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68 | (6) |
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2.3 The Damped Harmonic Oscillator |
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74 | (3) |
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2.4 Additional Analytic Methods for Special Systems |
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77 | (3) |
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2.5 Euler's Method for Systems |
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80 | (2) |
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2.6 Existence and Uniqueness for Systems |
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82 | (2) |
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2.7 The SIR Model of an Epidemic |
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84 | (2) |
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86 | (1) |
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87 | (4) |
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91 | (74) |
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3.1 Properties of Linear Systems and The Linearity Principle |
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92 | (6) |
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3.2 Straight-Line Solutions |
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98 | (11) |
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3.3 Phase Portraits for Linear Systems with Real Eigenvalues |
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109 | (11) |
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120 | (7) |
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3.5 Special Cases: Repeated and Zero Eigenvalues |
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127 | (9) |
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3.6 Second-Order Linear Equations |
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136 | (10) |
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3.7 The Trace-Determinant Plane |
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146 | (4) |
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3.8 Linear Systems in Three Dimensions |
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150 | (8) |
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158 | (7) |
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165 | (36) |
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4.1 Forced Harmonic Oscillators |
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166 | (13) |
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179 | (7) |
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4.3 Undamped Forcing and Resonance |
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186 | (6) |
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4.4 Amplitude and Phase of the Steady State |
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192 | (2) |
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4.5 The Tacoma Narrows Bridge |
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194 | (1) |
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195 | (6) |
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201 | (44) |
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5.1 Equilibrium Point Analysis |
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202 | (13) |
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215 | (6) |
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221 | (4) |
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225 | (7) |
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5.5 Nonlinear Systems in Three Dimensions |
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232 | (2) |
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5.6 Periodic Forcing of Nonlinear Systems and Chaos |
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234 | (2) |
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236 | (9) |
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245 | (40) |
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246 | (8) |
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6.2 Discontinuous Functions |
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254 | (6) |
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6.3 Second-Order Equations |
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260 | (10) |
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6.4 Delta Functions and Impulse Forcing |
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270 | (2) |
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272 | (4) |
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6.6 The Qualitative Theory of Laplace Transforms |
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276 | (1) |
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277 | (8) |
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285 | (20) |
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7.1 Numerical Error in Euler's Method |
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286 | (10) |
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7.2 Improving Euler's Method |
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296 | (4) |
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7.3 The Runge-Kutta Method |
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300 | (2) |
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7.4 The Effects of Finite Arithmetic |
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302 | (1) |
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302 | (3) |
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8 Discrete Dynamical Systems |
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305 | (12) |
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8.1 The Discrete Logistic Equation |
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306 | (2) |
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8.2 Fixed Points and Periodic Points |
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308 | (3) |
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311 | (2) |
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313 | (1) |
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314 | (3) |
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317 | |
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318 | (10) |
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B Power Series: The Ultimate Guess |
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328 | |
