quiz3sol1030

quiz3sol1030 - Since W { f, g } ( x ) = e-2 x cos( x ), it...

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TA: R McDougal Name: Math 415 25 October 2007 Quiz III – 10:30 Version Answer each question as completely as you can; remember you must show all work for full credit. You may not consult books, notes, or each other for this quiz. Good luck! Let f ( x ) = e - x . g ( x ) is some other function. Assume the Wronskian of f and g satisfies W { f, g } ( x ) = e - 2 x e ix + e - ix 2 , (1) where i 2 = - 1. 1. Use Euler’s Formula to simplify (1). Your final answer should not involve i or - 1. (3 points) W { f, g } ( x ) = e - 2 x e ix + e - ix 2 = e - 2 x cos( x ) + i sin( x ) + cos( - ix ) + i sin( - ix ) 2 = e - 2 x cos( x ) + i sin( x ) + cos( ix ) - i sin( ix ) 2 = e - 2 x cos( x ) . 2. Given that g (0) = 1, find g ( π/ 2). (7 points)
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Unformatted text preview: Since W { f, g } ( x ) = e-2 x cos( x ), it follows that fg-f g = e-2 x cos x . Thus e-x g + e-x g = e-2 x cos x . This is a rst order linear ODE, so after dividing by e-x to put the equation in the right form, we can solve it by using an integrating factor of = e x . Thus e x g + e x g = cos x , so d dx ( e x g ) = cos x . Thus g ( x ) = e-x (sin x + c ). Since g (0) = 1, c = 1. That is, g ( x ) = e-x (1 + sin x ) . Thus g ( / 2) = 2 e-/ 2 ....
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