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Unformatted text preview: More on Laplace Solution of Linear Initial Value Problems In this section we will be particularly concerned with non homogeneous linear initial value problems and their solution by Laplace transform methods. We will make use of both the stan dard procedure and the residue method , as convenient, to find the partial fractions decompositions of the Laplace trans forms y ( s ) of solutions as they are obtained. Example 1 Consider the problem d 4 y dx 4 5 d 2 y dx 2 + 4 y = sinh 3 x, y (0) = 1 , y (0) = 0 , y 00 (0) = 1 , y 000 (0) = 0 . Applying the Laplace transform we have s 4 ( L y ) ( s ) s 3 y (0) s 2 y (0) s y 00 (0) y 000 (0) 5 ( s 2 ( L y ) ( s ) s y (0) y (0)) + 4 ( L y ) ( s ) = 3 s 2 9 . Using the initial data, transposing terms and dividing the equa tion by s 4 5 s 2 + 4 we obtain ( L y ) ( s ) = 3 ( s 4 5 s 2 + 4)( s 2 9) + s ( s 2 4) s 4 5 s 2 + 4 = 3 ( s 2 1)( s 2 4)( s 2 9) + s s 2 1 ....
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 Spring '10
 ROBINSON

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