m2k_opm_slviv1

m2k_opm_slviv1 - More on Laplace Solution of Linear Initial...

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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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m2k_opm_slviv1 - More on Laplace Solution of Linear Initial...

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