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hmwk8

# hmwk8 - Mathematica to make a graph of the x-y the solution...

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Department of Chemical Engineering University of California, Santa Barbara Eng 5A Fall 2001 Instructor: David Pine Homework #8 Homework due: Wednesday, 28 November 2001 1. Consider the system of coupled differential equations described by the matrix equation: d x dt = Ax . (1) For each of the coefficient matrices A listed below, determine whether the origin for the matrix equation above is a sink, source, saddle, or center. Plot the direction field for each using Mathematica . In cases where the solution consists of real eigenvectors, show the eigendirections on your direction field plot. (a) A = 0 3 6 0 (b) A = - 2 1 1 2 (c) A = 0 1 2 - 2 2 (d) A = - 6 1 - 9 - 6 (e) A = - 1 1 - 5 1 (f) A = 2 1 3 - 3 - 2 (g) A = - 1 1 - 5 1 (h) A = - 2 4 - 1 0 2. Consider the coupled linear first order linear differential equations x 0 ( t ) = - 1 4 x ( t ) + 2 y ( t ) (2) y 0 ( t ) = - 2 x ( t ) - 1 4 y ( t ) , (3) subject to the initial conditions x (0) = 0 y (0) = 1 . (a) Use the DSolve function of Mathematica to find the solutions for x ( t ) and y ( t ). Then use Mathematica to plot these solutions as a function of t .

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(b) Use the ParametricPlot function of Mathematica
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Unformatted text preview: Mathematica to make a graph of the x-y the solution trajectory. You should obtain a spiral solution. Next, use Mathematica to generate a direction (or slope) ﬁeld plot of x vs. y (recall how you did something very similar to this in Homework #2). Finally, overlay the two plots so that you can see how the trajectory threads the direction ﬁeld plot. Describe in words how the trajectory would change if we have diﬀerent initial conditions from those given above. Do you expect all trajectories to have the same end point? (c) Rewrite equations (1) and (2) as a single equation in matrix form. Then use Mathematica to ﬁnd the eigenvalues and eigenvectors of the coeﬃcient matrix that appears in your equation. What do your results tell you about the nature of the solutions to equations (1) and (2)? Are they consistent with the particular solutions you found in part (a)?...
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