math1005test2sol

math1005test2sol - 40 marks-50 minutes MATH1005A,...

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MATH1005A, Summer2009 Test2 Problem 1 . Use the method of undetermined coefficients or some modification of it to find general solutions of the following differential equations. 1. ( 10 marks) y 00 - y 0 = xe 2 x . 2. ( 10 marks) y 00 + 4 y = sin 2 x. Problem 2 ( 10 marks) . Solve the differential equation x 2 y 00 + 5 xy 0 + 4 y = 0 , x > 0 . Problem 3 ( 10 marks) . Find a particular solution for the equation y 00 + 4 y = 3 csc x, using the method of variation of parameters. (you may need to use the identities, sin 2 x = 2 sin x cos x , cos 2 x = 1 - 2 sin 2 x and R csc xdx = ln | csc x - cot x | + c. ) Solution of 1.1: Auxiliary equation is r 2 - r = 0. Thus r 1 = 0, r 2 = 1. Thus y c = c 1 + c 2 e x . To find a y p we try y = ( Ax + B ) e 2 x . Then y 0 = e 2 x (2 Ax + 2 B + A ) , and y 00 = e 2 x (4 Ax + 4 B + 4 A ) . Plug these in the equation to get e 2 x (2 AX + (2 B + 3 A )) = xe 2 x . 1
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math1005test2sol - 40 marks-50 minutes MATH1005A,...

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