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Practice_MT_2_Math_33B_F08

Practice_MT_2_Math_33B_F08 - Mathematics 33B Practice...

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Mathematics 33B - Practice Midterm 2 Official exam to be administered Nov. 21st, 11:00am NAME (please print legibly): Your University ID Number: Your Discussion Section and TA: Signature: QUESTION VALUE SCORE 1 25 2 25 3 25 4 25 TOTAL 100 1
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1. (25 points) (a) Find the general solution to y 00 + y 0 - 6 y = 0. (b) Find a particular solution to y 00 + y 0 - 6 y = 4 te t . 2
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(c) Solve the initial value problem y 00 + y 0 - 6 y = 4 te t , y (0) = 1 / 4, y 0 (0) = - 3 / 4. 3
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2. (25 points) (a) Find the general solution to y 00 + 2 y 0 + 5 y = 0. 4
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(b) Find the general solution to y 00 + 2 y 0 + 5 y = 2 cos t . (c) As t → ∞ all solutions to y 00 + 2 y 0 + 5 y = 2 cos t approach the steady state solution. Write the steady state solution in the form A cos( t - δ ). You do not need to find δ explicitly, but do find tan δ and say which of the intervals ( - π, - π/ 2] , ( - π/ 2 , 0] , (0 , π/ 2], or ( π/ 2 , π ] contains δ . 5
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3. (25 points) This is a problem about the equations y 00 + p ( t ) y 0 + q ( t ) y = f ( t ) ( A ) and y 00 + p ( t ) y 0 + q ( t ) y = 0 , ( B ) where f ( t ) is not zero. Assume that y 1 and y 2 are solutions to (A) and that y 3 and y 4 are solutions to (B). For each linear combination of these functions below state whether it
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