pset_2sol

pset_2sol - ACM 95/100b Problem Set 2 Solutions Faisal...

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Unformatted text preview: ACM 95/100b Problem Set 2 Solutions Faisal Amlani January 24, 2009 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 SOLUTION : We first solve the homogeneous equation y 5 y + 6 y = 0 . Assume the form y = exp( x ) for the solution to the homogeneous equation (i.e. characteristic equation) = 2 5 + 6 = 0 . Clearly, solutions to this are 1 = 2 , 2 = 3 . Thus, two solutions to the homogeneous ODE are (ignore the constants without loss of generality): y 1 ( x ) = exp(2 x ) , y 2 ( x ) = exp(3 x ) . So want to find a particular solution y p such that the solution to the inhomogeneous ODE is given as y ( x ) = c 1 y 1 ( x ) + c 2 y 2 ( x ) + y p ( x ) for y 1 = exp(2 x ) , y 2 = exp(3 x ) solutions to the homogeneous ODE. By variation of parameters, we want y p as y p ( x ) = u 1 ( x ) y 1 ( x ) + u 2 ( x ) y 2 ( x ) . 1 We then have that (from class) that the solution for the inhomogeneous right hand side r ( x ) = cos( x ) y p ( x ) = y 1 ( x ) integraldisplay x y 2 ( t ) r ( t ) / W(t) dt + y 2 ( x ) integraldisplay x y 1 ( t ) r ( t ) / W(t) dt where the Wronskian W is given by W(t) = det vextendsingle vextendsingle vextendsingle vextendsingle y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ) vextendsingle vextendsingle vextendsingle vextendsingle = det vextendsingle vextendsingle vextendsingle vextendsingle exp(2 x ) exp(3 x ) 2 exp(2 x ) 3 exp(3 x ) vextendsingle vextendsingle vextendsingle vextendsingle = exp(5 x ) . Thus, plugging this and y 1 = exp(2 x ) , y 2 = exp(3 x ) , r ( t ) = cos( t ) we can integrate by parts to find that y p ( x ) = exp(2 x ) integraldisplay x exp( 2 t ) cos( t ) dt + exp(3 x ) integraldisplay x exp( 3 t ) cos( t ) dt = exp(2 x ) bracketleftbigg 2 5 exp( 2 x ) cos( x ) + 1 5 exp( 2 x ) sin( x ) bracketrightbigg + + exp(3 x ) bracketleftbigg 3 10 exp( 3 x ) cos( x ) + 1 10 exp( 3 x ) sin( x ) bracketrightbigg...
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pset_2sol - ACM 95/100b Problem Set 2 Solutions Faisal...

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