homework set 19-Solutions - B ˙ ILKENT UNIVERSITY...

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Unformatted text preview: B ˙ ILKENT UNIVERSITY Department of Mathematics MATH 225, LINEAR ALGEBRA and DIFFERENTIAL EQUATIONS, Solution of Homework set 1 # 19 U. Mu˘gan July 16, 2008 1) a) Corresponding homogenous equation is y 00- y- 2 y = 0 Since, it is a constant coefficient D.E. we look for the solution of the form y = e rx , then its C.E. is r 2- r- 2 = 0 with the roots r 1 = 2 and r 2 =- 1. Hence the homogenous solution is y h ( x ) = c 1 e 2 x + c 2 e- x , c 1 ,c 2 = constant . Assume y p = c 1 ( x ) e 2 x + c 2 ( x ) e- x then y p = [2 c 1 ( x ) e 2 x- c 2 ( x ) e- x ] + [ c 1 ( x ) e 2 x + c 2 ( x ) e- x ] . We set c 1 ( x ) e 2 x + c 2 ( x ) e- x . Computing y 00 p and substituting in the D.E. gives 2 c 1 ( x ) e 2 x- c 2 ( x ) e- x = 2 e- x . Thus we have two algebraic equations for c 1 ( x ) and c 2 ( x ) with the solution c 1 ( x ) = (2 / 3) e- 3 x , c 2 ( x ) =- 2 / 3 Hence c 1 ( x ) =- (2 / 9) e- 3 x and c 1 ( x ) =- (2 x/ 3). Substituting c 1 ( x ) and c 2 ( x ) in y p gives y p ( x ) =- (2 x/ 3) e- x . b) Similarly, first find the homogenous solution y h ( x ) = c 1 e- x + c 2 xe- x c 1 ,c 2 = constant . Assume y p = c 1 ( x ) e- x + c 2 ( x ) xe- x . c 1 ( x ) and c 2 ( x ) satisfy c 1 ( x ) e- x + c 2 ( x ) xe- x = 0 , and- c 1 ( x ) e- x + c 2 ( x )[ e- x- xe- x ] = 3 e- x with the solutions c 1 ( x ) =- 3 x, and c 2 ( x ) = 3 ....
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This note was uploaded on 06/18/2010 for the course COMPUTER S Math 225 taught by Professor Yosumhoca during the Spring '10 term at Bilkent University.

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homework set 19-Solutions - B ˙ ILKENT UNIVERSITY...

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