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4 - laplace_disc_ps

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

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Laplace Transform –Step functions – PS April 9, 2011 Problem 1 Find the Laplace transform of g ( t ) = ( t 2 0 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

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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. Let’s 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 - 1 3 / 2 ( s + 1 2 ) 2 - 9 4 = 2 3 e - 1 2 t sinh 3 2 t = 2 3 e - 1 2 t e 3 2 t - e - 3 2 t 2 !
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4 - laplace_disc_ps - Laplace Transform Step functions PS...

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