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pset_2

# pset_2 - ξ = ln x may be of some help(b Over what...

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ACM 95/100b Problem Set 2 January 19, 2009 Due by 5:00PM on 1/23/2009 Please deposit your problem set in the slot in 303 Firestone or upload it via Moodle. In either case please keep a copy of your problem set. Please remember to include your section number and section instructor. Each problem is worth 10 points - each part of a multi-part problem is weighted equally. Collaboration is allowed on all problems but please write up the solutions yourself. Problem 1 Solve the following second order ODE using variation of parameters y ′′ - 5 y + 6 y = cos( x ) Using your general solution, determine the solution to the initial value problem y (0) = 1 y (0) = 0 Problem 2 Consider the solution of the second order ODE y ′′ + y x - 9 y x 2 = 0 . (a) Obtain the general solution of this ODE. The substitution ξ = ln x may be of

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Unformatted text preview: ξ = ln x may be of some help. (b) Over what intervals would a solution to the initial value problem exist and be unique? (c) Show that the following initial value problem has no solution y (0) = 1 y ′ (0) = 2 1 (d) Show however that the following problem has a solution but it’s not unique. y (0) bounded Explain these results in the context of the existence and uniqueness theorem. Problem 3 Solve the following system of equations x ′ = 4 2-2-5 3 2-2 4 1 x , x (0) = 3 4 . Problem 4 Using integration find the Laplace transform of (a) cos az , (b) exp( az ) cos bz , (c) z exp( az ) sin bz . Problem 5 Use the method of Laplace transforms to solve the system x ′ = p 4 1-1 2 P x x (0) = x 2...
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