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quiz8 and ans

quiz8 and ans - y 1 Answer f t y = 10 y 2-20 P y P = y(0 1...

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AMSC/CMSC 660 Quiz 8 , Fall 2006 1. (10) Let i = - 1, and suppose we have a system of differential equations y 0 = y ( t, y ) with 3 components. Suppose the system has a Jacobian matrix J ( t, y ) with eigenvalues 4 - t 2 , - t - it, - t + it. For what values of t is the equation stable? Answer: We need the real parts of all eigenvalues to be negative. This means 4 - t 2 < 0 and - t < 0, so the equation is stable when t > 2. 2. Let y 0 = 10 y 2 - 20 , y (0) = 1 . Apply a PECE scheme to this problem, using Euler and Backward Euler with a stepsize h = . 1, to obtain an approximation for
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Unformatted text preview: y ( . 1). Answer: f ( t, y ) = 10 y 2-20. P: y P = y (0) + . 1 f (0 , y (0)) = 1 + . 1(-10) = 0 . E: f P = f ( . 1 , y P ) = 10 *-20 =-20 . C: y C = y (0) + . 1 f P = 1-2 =-1 . E: f C = f ( . 1 , y C ) = 10-20 =-10 . Note that the predicted and corrected values are quite dierent, so neither can be trusted; we should reduce the stepsize and recompute. The true value is y ( . 1) -. 69. 1...
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