wksht07sol - MATH 152 L1 & L2 Applied Linear...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Spring 2004-05 ¥ Week 07 Worksheet Solutions : Laplace Transform Q1. (Inverse Laplace transform) Find the inverse Laplace transform of F ( s ) = e- π 3 s s 2 + 1 . [ Hint . You may use the formula L- 1 { e- as G ( s ) } = u ( t- a ) g ( t- a ), where G ( s ) = L{ g ( t ) } .] Solution Because L- 1 ‰ 1 s 2 + 1 = sin t , then by the given formula, we have L- 1 ‰ e- π 3 s s 2 + 1 = u ( t- π 3 ) sin( t- π 3 ). Q2. (Impulse function) Use Laplace transform to solve the boundary value problem (BVP) y 00 + y = 2 δ ( t- π 3 ) , 6 t 6 π 2 , y (0) = 0 , y ( π 2 ) = 1 2 . [ Hint . You may first assume that y (0) = A . Then use the Laplace method to solve the initial value problem. Finally you can determine the constant A by making use of the boundary condition.] Solution Let y (0) = A . Then consider the following initial value problem y 00 + y = 2 δ ( t- π 3 ) , y (0) = 0 , y (0) = A.A....
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wksht07sol - MATH 152 L1 & L2 Applied Linear...

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