219 hw3and solution from 2005

Elementary Differential Equations, with ODE Architect CD

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Math 219, Homework 3 Due date: 9.12.2005, Friday 1. Consider the initial value problem d 2 x dt 2 + dx dt + x = u 4 ( t ) , y (0) = y 0 (0) = 0 (a) Solve this initial value problem using the Laplace transform. (b) Use ODE Architect to solve the equation, and graph the solution. Also graph dx dt with respect to t (You can use the function Step ( t, 4) to create a unit step function with discontinuity at t = 4). (c) Discuss how the graphs agree with the solutions in (a): in particular determine (if any) all the points where x ( t ) and dx dt are discontinuous, behavior of these two functions for t → ∞ , their maxima and minima. 2. Write each of the following systems of differential equations in matrix form, find the eigenvalues and eigenvectors of the coefficient matrices, and using these, find
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Unformatted text preview: all solutions of each system. Also, graph the phase portraits ( x-y graph) using ODE Architect. Please use a scale which includes the point (0 , 0), and graph several solutions in order to clearly observe the behavior around (0 , 0). Also, place arrows on the solution curves which indicate the direction of increasing t , and make sure that solution curves along the eigenvector directions are graphed if there are any real eigenvectors. (a) dx dt = 2 x-y dy dt = 3 x + 3 y (b) dx dt =-x + y dy dt = 3 x-4 y (c) dx dt = 2 x + 3 y dy dt = 5 x + 5 y (d) dx dt =-4 x + 3 y dy dt =-3 x + 2 y (e) dx dt =-x-3 y dy dt = 2 x + y...
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