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**Unformatted text preview: **y ( t ) to (1) is given by (5) y ( t ) = ( y I ( t ) if 0 ≤ t < a, y II ( t ) if t ≥ a. On the other hand, the application of the Laplace transform to the initial value problem (1) has the advantage of solving the problem in Date : February 22, 2011. 1 2 KIAM HEONG KWA a more direct manner. Let Y ( s ) = L y ( s ). Then taking the Laplace transform of (1) yields ( s 2 + αs + β ) Y ( s )-( s + α ) y-y = e-as s , from which it follows that (6) Y ( s ) = ( s + α ) y + y s 2 + αs + β + e-as s ( s 2 + αs + β ) . The solution y ( t ) to (1) is then obtained by inverting the transform: (7) y ( t ) = L-1 [ Y ( s )]( t ) = L-1 ± ( s + α ) y + y s 2 + αs + β + e-as s ( s 2 + αs + β ) ² ( t ) ....

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