The Computable Differential Equation Lecture ... - Bruce E. Shapiro

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16 CHAPTER 1. CLASSIFYING THE PROBLEM {{ y[t] → e 3t2 }} 2 C[1] (1.86) This means that Mathematica has determined that there is one solution to the ODE and it has the form y(t) = C 1 e 3t2 /2 , where C 1 is an arbitrary constant. Because no initial condition is specified, the system defines an arbitrary constant C[1]. Subsequent constants, if needed, are added as C[i]. For example, the second order equation y ′′ = 3ty (1.87) is a form of Airy’s equation (compare it with the previous example). The standard way of writing the solution to this is ( ) ( ) y(t) = C 1 Ai 3 1/3 t + C 2 Bi 3 1/3 t (1.88) where Ai and Bi are the Airy functions of the first and second kind. To solve this in Mathematica, we enter DSolve[y’’[t]==3 t y[t], y[t], t] (1.89) and the system responds {{ y[t] → AiryAi[3 1/3 t]C[1]+AiryBi[3 1/3 }} t]C[2] (1.90) To solve a systems of equations, such as y ′ = 3x (1.91) x ′ = y (1.92) y(0) = √ 3 (1.93) x(0) = 1 (1.94) one would type in DSolve[ {y’[t] == 3 x[t], x’[t] == y[t], y[0] == Sqrt[3], x[0] == 1 }, {x[t], y[t]}, t] and obtain the solution {{ y[t] → √ √ √ }} 3e 3 t, x[t] → e 3 t (1.95) Numerical Solutions of ODEs with Mathematica Mathematica has a number of built in algorithms to solve differential equations numerically; the function that does this is NDSolve. To solve the systems of equations y ′ = −x − .1y (1.96) x ′ = 3y (1.97) y(0) = √ 3 (1.98) x(0) = 1 (1.99) Math 582B, Spring 2007 California State University Northridge c○2007, B.E.Shapiro Last revised: May 23, 2007
CHAPTER 1. CLASSIFYING THE PROBLEM 17 for 0 ≤ t ≤ 100, the command is ns = NDSolve[{y’[t]==-x[t]-.1y[t], x’[t]==3y[t], y[0]==Sqrt[3], x[0]==1 }, {x[t], y[t]}, {t, 0, 1000}] The solution is returned is {{x[t] → InterpolatingFunction[{{0.,1000.}}, ][t] y[t] → InterpolatingFunction[{{0.,1000.}}, ][t]}} The interpolating functions can be treated like any other function. For example, to plot the solution for x(t) from t = 0 to t = 25, Plot[Evaluate[x[t] /. ns], t, 0, 25] To plot both x(t) and y(t) from t = 0 to t = 100, with x plotted in red, and y plotted in blue, ⇐= Plot[Evaluate[{x[t], y[t]} /. ns], {t, 0, 100}, PlotStyle -> {Red, Blue}, PlotRange -> All] c○2007, B.E.Shapiro Last revised: May 23, 2007 Math 582B, Spring 2007 California State University Northridge
Page 1 and 2: The Computable Differential Equatio
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Page 5 and 6: CONTENTS v Timeline on Computable D
Page 7 and 8: Chapter 1 Classifying The Problem 1
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CHAPTER 3. APPROXIMATE SOLUTIONS 67
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Chapter 4 Improving on Euler’s Me
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CHAPTER 4. IMPROVING ON EULER’S M
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Chapter 5 Runge-Kutta Methods 5.1 T
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CHAPTER 5. RUNGE-KUTTA METHODS 91 w
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CHAPTER 5. RUNGE-KUTTA METHODS 93 w
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CHAPTER 5. RUNGE-KUTTA METHODS 95 5
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Chapter 6 Linear Multistep Methods
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CHAPTER 6. LINEAR MULTISTEP METHODS
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Chapter 7 Delay Differential Equati
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CHAPTER 7. DELAY DIFFERENTIAL EQUAT
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Chapter 8 Boundary Value Problems 8
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CHAPTER 8. BOUNDARY VALUE PROBLEMS
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Chapter 9 Differential Algebraic Eq
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CHAPTER 9. DIFFERENTIAL ALGEBRAIC E
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Chapter 10 Appendix on Analytic Met
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Bibliography [1] Ascher, Uri M., Ma
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