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MIT 18 03 - Exam II

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18.03 Hour Exam II March 17, 2010 Your Name Your Recitation Leader’s Name Your Recitation Time Problem Points 1 2 3 4 5 Total Do not open this booklet till told to do so. There are five problems. Use your test-taking skills—be sure you get to all the problems. Do all your work on these pages. No calculators or notes may be used. The point value (out of 100) of each problem is marked in the margin. Solutions w ill be available on the web after 4:00 today, and at recitation.[8] 1. (a) For what value of k is the system represented by ¨x+ ˙x+kx = 0 critically damped? [4] (b) For k greater than that value, is the system overdamped or underdamped? [8] (c) Suppose a solution of x¨ + ˙x + kx = 0 vanishes at t = 1, and then again for t = 2 (but not in between). What is k?[10] 2. (a) Find a solution of x¨ + x = 5te2t . [10] (b) Suppose that y(t) is a solution of the same equation, ¨x+x = 5te2t, such that y(0) = 1 and ˙y(0) = 2. (This is probably not the solution you found in (a).) Use y(t) and other functions to write down a solution x(t) such that x(0) = 3 and ˙x(0) = 5.[10] 3. (a) Consider the equation x¨ + bx˙ + kx = cos(ωt). We will vary the spring constant but keep b fixed. For what value of k is the amplitude of the sinusoidal solution of x¨ + bx˙ + kx = cos(ωt) maximal? (Your answer will be a function of ω and may depend upon b as well.) d3x dx [10] (b) (Unrelated to the above.) Find the general solution of = 0. dt3 − dt4. A certain system has input signal y and system response x related by the differential equation x¨ + ˙x + 6x = 6y. It is subjected to a sinusoidal input signal. [10] (a) Calculate the complex gain H(ω). [5] (b) Compute the gain at ω = 2. [5] (c) Compute the phase lag at ω = 2.5. Suppose that 12 t sin(2t) is a solution to a certain equation mx¨ + bx˙ + kx = 4 cos(2t). [4] (a) Write down a solution to mx¨ + bx˙ + kx = 4 cos(2t − 1). [4] (b) Write down a solution to mx¨ + bx˙ + kx = 8 cos(2t). [12] (c) Determine m, b, and k.MIT OpenCourseWare http://ocw.mit.edu 18.03 Differential Equations���� Spring 2010 For information about citing these materials or our Terms of Use, visit:


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MIT 18 03 - Exam II

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