Computer Algebra Recipes

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4.1. FIRST-ORDER MODELS 163 μ tg v := tanh V V To determine y(t), a second ODE is formed by equating dy=dt to v. To make the ¯nal expression appear in the desired form, the convert command is used to reexpress the hyperbolic tangent in terms of the hyperbolic sine and cosine. > ode2:=diff(y(t),t)=convert(v,sincos); ode2 := d μ tg sinh V V y(t) = μ dt tg cosh V Analytically solving ode2 for y(t), subject to the initial condition y(0) = 0, > dsolve(fode2,y(0)=0g,y(t)); V y(t) = 2 μ μ tg ln cosh V g produces an output that is identical in structure to equation (4.4), thus completing Vectoria's task. PROBLEMS: Problem 4-15: Return velocity A ball of unit mass is thrown vertically upward near the earth's surface with an initial speed v(0) = U. Assuming that Newton's law of air resistance prevails, show that after the ball rises to its maximum height and begins to fall it passes its initial position with a velocity v = (UV)=( p U 2 + V 2 ); where V is the terminal velocity. Hint: Reexpress the acceleration as dv dt = μ μ μ dv dx dv(x) = v(x) dx dt dx and note that at the maximum height the speed is zero. Problem 4-16: A Riccati equation Consider the following Riccati equation: y 0 (x)+y(x)=x + ay(x) 2 + b =0; with the initial condition y(0) = A, wherea, b, andAare real constants. (a) Analytically solve the ODE. Does the answer depend on the value of A? (b) What does the general solution look like if no initial condition is speci¯ed? Explain what happens mathematically when y(0) = A is imposed. (c) Taking a =5andb =2,plotthesolutionfory(0) = A over the range x = 0 to 2, using the plot option view=[0..2,-5..5]. (d) Explain the origin of the singular points in the graph in terms of the behavior of Bessel functions.
164 CHAPTER 4. NONLINEAR ODE MODELS Problem 4-17: Nonlinear drag on Lake Ogopogo A boat is launched on Lake Ogopogo with initial speed v0. Thewaterexertsa drag force F (v) =¡aebv ,witha>0, b>0, thus slowing the boat down. (a) Find an analytic expression for the speed v(t). (b) Determine the time it takes for the boat to come to rest. (c) How far does the boat travel along Lake Ogopogo before coming to rest? 4.2 Second-Order Models The vast majority of second-order nonlinear models that physicists and engineers are interested in must be solved numerically. However, we begin this section with a few examples of models that lead to nonlinear ODEs having exact analytic solutions. Historically, these models have appeared in many equivalent guises, so keep this in mind when you read the stories. 4.2.1 Patches Gives Chase Man ... cannot learn to forget, but hangs on the past: however far or fast he runs, that chain runs with him. Friedrich Nietzsche, German philosopher (1844{1900) A loveable beagle, named Patches, is patrolling a °at farm ¯eld, sni±ng contentedly at gopher holes, when she spots her mistress, Heather, walking at constant speed along a straight road at the edge of the ¯eld. Patches then runs at constant speed toward Heather in such a way as to aim always at her with her sensitive beagle nose. With distances in km, Patches is initially at (x =1;y= 0) and Heather at (0; 0). The road is described by the equation x = 0, and the ratio of Heather's speed to Patches' speed is r. (a) Derive the nonlinear ODE describing Patches' path y(x). (b) Analytically solve the ODE for y(x). If Heather walks at 3 km/h and Patches runs at 8 km/h, plot y(x) using constrained scaling. (c) How many minutes does it take the dog to reach her mistress? How far has Heather walked in this time? How far has Patches run? (d) Animate the motion of Patches and Heather, including a tangent line to Patches' instantaneous position, which points towards Heather. Again, use constrained scaling. At some instant in time, Patches' coordinates are (x; y(x)), while Heather's are (0; h). Patches runs towards Heather in such a way as to aim always at her. Therefore the slope of the tangent to Patches' path is dy=dx =(h¡y(x)=(0¡x), whichisnowenteredineq.
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Richard H. Enns George C. McGuire C
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PREFACE A computer algebra system (
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viii CONTENTS 2.3.1 Finite Di®eren
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x CONTENTS 8.3 The Bifurcation Diag
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2 INTRODUCTION B. Computer Algebra
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4 INTRODUCTION (a) At what angle Á
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6 INTRODUCTION The negative answer
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8 INTRODUCTION D. Maple Help We tea
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Part I THE APPETIZERS A man ceases
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14 CHAPTER 1. PHASE-PLANE PORTRAITS
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48 CHAPTER 2. PHASE-PLANE ANALYSIS
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Chapter 3 Linear ODE Models Among a
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Page 153 and 154: 150 CHAPTER 4. NONLINEAR ODE MODELS
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214 CHAPTER 5. LINEAR PDE MODELS. P
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Chapter 6 Linear PDE Models. Part 2
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6.1. WAVE EQUATION MODELS 249 > sol
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6.1. WAVE EQUATION MODELS 251 6.1.2
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6.1. WAVE EQUATION MODELS 253 The p
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6.1. WAVE EQUATION MODELS 255 Recog
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6.1. WAVE EQUATION MODELS 257 PROBL
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6.1. WAVE EQUATION MODELS 259 Then
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6.2. SEMI-INFINITE AND INFINITE DOM
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6.3. NUMERICAL SIMULATION OF PDES 2
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Part III THE DESSERTS The way a chi
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288 CHAPTER 7. THE HUNT FOR SOLITON
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320 CHAPTER 8. NONLINEAR DIAGNOSTIC
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356 BIBLIOGRAPHY [Dav62] H. T. Davi
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358 BIBLIOGRAPHY [Nyq28] H. Nyquist
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Index acoustical waveguide, 261 ade
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INDEX 363 eigenvalue, 130 Eiram Eir
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INDEX 365 Help, Full Text Search, 8
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INDEX 367 laplace, 121,267,268 lhs,
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INDEX 369 ¯sh harvesting, 100 ¯xe
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INDEX 371 quartic map, 345 Queen Di
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Computer Algebra Recipes

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