20d_07s_fe

20d_07s_fe - (b Find the ²rst six terms(i.e up to and including the coe³cient of x 5 of the particular solution that satis²es the initial

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Math 20D Final Exam 100 points June 12, 2007 Print Name and ID number on your blue book. BOOKS and CALCULATORS are NOT allowed. Both sides of one page of NOTES is allowed. You must show your work to receive credit. 1. (60 pts.) Find the solution for each of the following di±erential equations. If initial conditions are given, ²nd the particular solution. If no initial conditions are given, ²nd the general solution. DO NOT LEAVE INTEGRALS IN YOUR ANSWERS. DO NOT LEAVE COMPLEX NUMBERS IN YOUR ANSWERS. (a) y + ty = t 3 with y (0) = 0. (b) t 2 y + e y = 0 on the interval t > 0. (c) y ′′ - 2 y + 2 y = 0. (d) y ′′ - 4 y + 4 y = 2 e t . (e) The system of two equations x 1 = 2 x 1 - x 2 and x 2 = 3 x 1 - 2 x 2 . 2. (18 pts.) This exercise deals with the power series solution for the di±erential equation y ′′ - xy - y = 0 about x 0 = 0. (a) Find the recurrence relation for the coe³cients of the power series solution.
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Unformatted text preview: (b) Find the ²rst six terms (i.e., up to and including the coe³cient of x 5 ) of the particular solution that satis²es the initial conditions y (0) = 1 and y ′ (0) = 2. 3. (12 pts.) The motion of a simple rigid pendulum without friction can be described by the di±erential equation d 2 θ/dt 2 =-K sin θ where K > 0 and θ is the angle the pendulum makes with the downward vertical. (a) Let ω = dθ/dt . Show that d 2 θ/dt 2 = ω dω/dθ . (b) It follows from (a) that the pendulum equation can be written ω dω/dθ =-K sin θ . Solve this equation. 4. (10 pts.) Given that y ′′-y = g ( t ), y (0) = 1, y ′ (0) = 2 and g ( t ) = b 1-t for 0 ≤ t ≤ 1 for t ≥ 1, ²nd Y ( s ), the Laplace transform of y ( t ). END OF EXAM...
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This note was uploaded on 06/18/2009 for the course MATH 20D taught by Professor Mohanty during the Fall '06 term at UCSD.

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