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# m238sp05t4s - Mathematics 238 test four solutions Friday...

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Mathematics 238 test four solutions Friday, June 10, 2005 please tell me about any errors 1 . Find the function which corresponds to each Laplace transform: ( a ) 2 ( s + 4) 3 apply the translation principle: function transform t 2 2! s 3 = 2 s 3 t 2 e - 4 t 2 ( s + 4) 3 Therefore L - 1 2 ( s + 4) 3 = t 2 e - 4 t ( b ) L - 1 s + 3 s 2 - s - 6 = L - 1 s + 3 ( s - 3)( s + 2) partial fractions = L - 1 6 5 s - 3 + - 1 5 s + 2 = 6 5 e 3 t - 1 5 e - 2 t ( c ) 6 e - 2 s s 2 + 9 apply the 2 nd translation principle: function transform 2 sin 3 t 2 · 3 s 2 + 3 2 u 2 ( t ) (2 sin 3( t - 2)) 2 · 3 s 2 + 3 2 · e - 2 s Therefore L - 1 6 e - 2 s s 2 + 9 = 2 u 2 ( t ) sin 3( t - 2) 2 . Simplify each of the following: ( a ) 0 e - 3 t cos 4 t e - st dt = L e - 3 t cos 4 t = s + 3 ( s + 3) 2 + 4 2 ( b ) Use the convolution theorem L t 2 * t 3 = L t 2 ·L t 3 = 2 s 3 · 6 s 4 = 12 s 7 = L 1 60 t 6 Therefore t 2 * t 3 = 1 60 t 6

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page two 3 . Sketch the graph of the function f ( t ) = 3( u 2 ( t ) - u 4 ( t )) + (7 - t )( u 4 ( t ) - u 7 ( t )) = 0 if t < 2 3 if 2 t < 4 7 - t if 4 t < 7 0 if t 7 t y 1 2 3 4 5 6 7 1 2 3 4 . Given the function g ( t ) = 0 if t < 0 1 - t if 0 t < 2 t - 3 if 2 t < 3 0 if t 3 Rewrite the function in terms of the unit functions u c ( t ) g ( t ) = (1 - u 2 ( t ))(1 - t ) + ( u 2 ( t ) - u 3 ( t ))( t - 3) Calculate the Laplace transform of this function. L { g ( t ) } = L { (1 -
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