MATH 376: Ordinary Differential Equations SAMPLE MIDTERM II Fall 2012 NAME : NOTE : There are 5 problems on this midterm (total of 6 pages). Use of calculators to check your work is permitted; however, in order to receive full credit for any problem, you must show work leading to your answer. You have 75 minutes to complete this test. Problem Possible points 1 20 2 15 3 30 4 15 5 20 Total 100 Score MATH 376 – SAMPLE MIDTERM II Problem 1. (20pts) Solve the initial value problem. y 00 − 6y 0 + 13y = 0, y(0) = 0, y 0 (0) = 1 Page 2 MATH 376 – SAMPLE MIDTERM II Page 3 Problem 2. (15pts) Set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients. y 00 − 6y 0 + 13y = xe3x sin 2x MATH 376 – SAMPLE MIDTERM II Page 4 Problem 3. (30pts) Find the general solution, then sketch a typical solution. 0 x 4 1 x 0 = y 6 −1 y BONUS(+5pts): For which initial conditions (x0 , y0 ) will the solution to the initial value problem satisfy limt→∞ (xt , yt ) = (0, 0). MATH 376 – SAMPLE MIDTERM II Page 5 Problem 4. (15pts) Transform the differential equation into an equivalent system of first-order differential equations and write your answer in matrix form. t3 x(3) − 2t2 x00 + 3tx0 + 5x = ln t MATH 376 – SAMPLE MIDTERM II Page 6 Problem 5. (20pts) Apply the Improved Euler’s method with a step size h = 0.2 to approximate the solution to the following initial value problem on the interval [0, .4]. y 0 = −2xy, y(0) = 2

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