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Unformatted text preview: Name: PID: TA: Sec. No: Sec. Time: Math 20D Midterm Examination July 14, 2010 Turn off and put away your cell phone. No calculators or any other devices are allowed on this exam. You may use one 3x5 card of notes, but no books or other assistance on this exam. Read each question carefully, answer each question completely, and show all of your work. Write your solutions clearly and legibly; no credit will be given for illegible solutions. If any question is not clear, ask for clarification. # Points Score 1 10 2 9 3 9 4 9 5 9 6 9 55 1. (a) (5 points) Solve the following initial value problem: y- y = e x cos x- 2 y (0) = 3 Solution: We use the method of integrating factors. We find the integrating factor is e R- 1 dx = e- x , so we obtain the differential equation e- x y- e- x y = cos x- 2 e- x . Integrating this equation gives e- x y = sin x + 2 e- x + C. We can then use the initial condition to solve for C to obtain C = 1. Solving the equation e- x y = sin x + 2 e- x + 1 for y gives y = e x sin x + 2 + e x . (b) (5 points) Solve the following initial value problem: dy/dx = xy 2 x 2 +2 y (1) =- 2 / ln(6) Solution: We use the method of separation of variables. We can rewrite the differential equation as y- 2 dy = x x 2 +2 dx . Integrating this equation gives- y- 1 = 1 2 ln( x 2 + 2) + C....
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- Spring '04