2 y c 1 e 3 t c 2 e t b find the solution to the

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y = c 1 e 3 t + c 2 e - t (b) Find the solution to the given initial value problem. y = 2 e 3 t + 5 e - t (c) Calculate the Wronskian for the fundamental set of solutions you found while solving part (a). W = det " e 3 t e - t 3 e 3 t - e - t # = - 4 e 2 t (d) Transform the given second order differential equation into an initial value problem involving a system of first order equations. x 0 1 = x 2 x 0 2 = 3 x 1 + 2 x 2 x 1 (0) = 7 x 2 (0) = 1 or ~x 0 = " 0 1 3 2 # ~x, ~x 0 (0) = " 7 1 # (e) Find the general solution to this system. " x 1 x 2 # = c 1 " 1 3 # e 3 t + c 2 " 1 - 1 # e - t (f) Solve the initial value problem you wrote in part (d). Does it agree with your answer in part (b)? Why or why not? " x 1 x 2 # = 2 " 1 3 # e 3 t + 5 " 1 - 1 # e - t 3
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Yes, y = x 1 has solution x 1 = 2 e 3 t + 5 e - t . (g) Calculate the Wronskian for the fundamental set of solutions you found while solving part (e). How does it compare to the Wronskian you found in part (b)? Equal. Or it is a multiple (this will occur if you picked a different eigenvector than we did, for example: " - 1 1 # for r = - 1, you get W = 4 e 2 t ). 6. Let ~x 0 = " 1 - 1 5 - 3 # ~x (a) Find the general solution. ~x = c 1 e - t " cos t 2 cos t + sin t # + c 2 e - t " sin t - cos t + 2 sin t # e - t , (b) What type of solution is the origin? Stable spiral (c) Sketch the phase plane, include several trajectories. Make sure your diagram agrees with your answer in part (b). (d) What is the long term behavior of the trajectories? Head towards ~ 0. 7. Let ~x 0 = " 2 - 1 3 - 2 # ~x (a) Find the general solution. ~x = c 1 " 1 1 # e t + c 2 " 1 3 # e - t 4
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(b) What type of solution is the origin? Saddle point (c) Sketch the phase plane, include several trajectories. Make sure your diagram agrees with your answer in part (b). 8. Let ~x 0 = " 3 - 4 1 - 1 # ~x (a) Find the general solution. Ans: ~x = c 1 " 2 1 # e t + c 2 " 2 1 # te t + " 1 0 # e t ! (b) What type of solution is the origin?
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