33B Mathematics - Midterm 2
Instructor : D. E. Weisbart February 27, 2009
NAME (please print legibly): Your University ID Number: Your Discussion Section and TA: Signature:
There are FIVE questions on this examination. Calculators, notes and books may not be used in this examination. Full credit may not be given if insucient work is shown.
QUESTION VALUE SCORE 1 2 3 4 5 TOTAL 20 20 20 20 20 100
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1. (20 points) a) Find the general solution of the equation x + 5x + 6x = 0.
b) Find the general solution of the equation x + 5x + 6x = 52 cos 2t. Make sure to show your work.
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2. (20 points) a) Find the general solution of the equation x 4x + 4x = 0.
b) Find the general solution of the equation x 4x + 4x = e2t . Make sure to show your work.
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3. (20 points) a) If an undamped mass-spring system starts at rest and is subjected to a force F (t) = 10et , the displacement of the mass at time t is a solution to the initial value problem x + 4x = 10et , x(0) = 0, x (0) = 0. Find x(t).
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b) Since the force in part a) goes to zero very quickly as t increases, the solution x(t) tends to a solution of the homogeneous equation x + 4x 0 = as t increases. So x(t) tends to A cos(kt ). Find A, k and .
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4. (20 points) The dierential equation x + x = 0 has two linearly independent solutions y1 = cos x and y2 = sin x. a. Compute the Wronskian, W (y1 , y2 ), of y1 and y2 .
b. Use variation of parameters to solve the equation x + x = sin2 t. You are expected to be able to perform the integrals in the problem as they are quite standard.
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5. (20 points) For each of the statements below, say whether the statement is True or False. Suppose that a, b and f are continuous functions. a) If y and z are linearly independent solutions to the equation x (t) + a(t)x (t) + b(t)x = 0, then y(0) and z(0) cannot both be 0.
b) The solutions to the equation x (t) + a(t)x (t) + b(t)x = f (t) exist for all time.
c) The graphs of distinct solutions to the equation x (t) + a(t)x (t) + b(t)x = f (t) can never cross.
d) If y1 (t) and y2 (t) are solutions to x (t) + a(t)x (t) + b(t)x(t) = f (t) and y1 (t) + y2 (t) is also a solution, then f (t) must be equal to 0.
e) Curves that are the graphs of solutions to second order dierential equations cannot run o to innity in nite time.
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