4 - laplace_disc_ps - Laplace Transform Step functions PS...

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Unformatted text preview: Laplace Transform Step functions PS April 9, 2011 Problem 1 Find the Laplace transform of g ( t ) = ( t 2 t < 2 6 2 < t < (1) Solution : g ( t ) in terms of the Heaviside function is: g ( t ) = t 2- u 2 ( t )( t 2- 6) . Evaluating the two separately, we write L t 2 = 2 s 3 . (2) For the second part we need to use the t-shifting theorem: Step 1: f ( t- 2) = t 2- 6 Step 2: f ( t ) = ( t + 2) 2- 6 = t 2 + 4 t- 2 Step 3: L u 2 ( t )( t 2- 6) = e- 2 s L t 2 + 4 t- 2 = e- 2 s 2 s 3 + 4 s 2- 2 s (3) Combining the results: L { g ( t ) } = 2 s 3 + e- 2 s 2 s 3 + 4 s 2- 2 s . (4) 1 c HF, 2011 Laplace Transform Step functions PS April 9, 2011 Problem 2 Find the inverse Laplace transform of F ( s ) = e- 2 s s 2 + s- 2 (5) Solution : We can either perform do partial fractions, or complete the square here. Lets do the latter: F ( s ) = e- 2 s ( s + 1 2 ) 2- 9 4 (6) For the second part we need to use the t-shifting theorem: Step 1: L- 1 1 ( s + 1 2 ) 2- 9 4 = 2 3 L...
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This note was uploaded on 05/02/2011 for the course GE 207K taught by Professor None during the Spring '10 term at University of Texas at Austin.

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4 - laplace_disc_ps - Laplace Transform Step functions PS...

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