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finals06 - 92.236 Engineering Dierential Equations Final...

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92.236 Engineering Differential Equations Final Exam Spring 2006 Please write all answers and all work in the blue book provided. PLEASE SHOW ALL WORK! You will not receive full credit if you do not show your work. Problem 1. (15 pts.) Solve the following initial value problem: xy + 5 y = 7 x 2 with y (2) = 5. Problem 2. (15 pts.) Solve the following initial value problem: xyy = x 2 + 3 y 2 with y (1) = 2. Problem 3. (10 points) Consider the first-order autonomous equation dx dt = 4 x - x 3 a. Draw the phase line for the given equation, noting the equilibrium solutions. b. Sketch the graphs of the five solutions that satisfy the five initial conditions x (0) = - 2 , x (0) = - 1 , x (0) = 0 , x (0) = 2 , x (0) = 3. c. If x (0) = 3, determine the limiting value of x ( t ) as t increases. Problem 4. (10 points) Find the general solution to each of the following differential equations: a. y + 4 y + 5 y = 0 b. y (4) - 9 y = 0 Problem 5. (15 points) Solve the following initial value problem: y - 2 y = 8 cos(2 x ) with y (0) = 1 and y (0) = 0 .
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