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# Sol-HW6 - Math Methods Fall 2010 Solutions for Assignment 6...

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Math Methods Solutions for Assignment 6 Oct. 11, 2010 Fall 2010 Laplace Transforms Due Oct. 18, 2010 1 (5). Find the Laplace transform of function f ( t ) = te t . Determine the region of existence of Laplace transform. L ( te t ) = Z 0 e - st t e t dt = Z 0 t e (1 - s ) t dt = 1 1 - s t e (1 - s ) t 0 - Z 0 1 1 - s e (1 - s ) t dt = = - 1 (1 - s ) 2 e (1 - s ) t 0 = 1 (1 - s ) 2 , for Re( s ) > 1. 2 (6). Find the inverse Laplace transform of a) 1 s 3 - s 2 1 s 3 - s 2 = 1 s - 1 - 1 s - 1 s 2 L - 1 1 s 3 - s 2 = L - 1 1 s - 1 - L - 1 1 s - L - 1 1 s 2 = e t - 1 - t b) s 2 + s - 1 s 3 - 2 s 2 + s - 2 s 2 + s - 1 s 3 - 2 s 2 + s - 2 = s 2 + s - 1 ( s - 2)( s - i )( s + i ) = 1 s - 2 - i/ 2 s - i + i/ 2 s + i = 1 s - 2 + 1 s 2 + 1 L - 1 s 2 + s - 1 s 3 - 2 s 2 + s - 2 = L - 1 1 s - 2 + 1 s 2 + 1 = e 2 t + sin t 3 (6). Solve the differential equations using Laplace transforms: a) y 0 + y = cos t , for t > 0 , y (0) = 1 . s ˆ y ( s ) - y (0) + ˆ y ( s ) = s s 2 + 1 ˆ y ( s ) = s ( s + 1)( s 2 + 1) + 1 s + 1 = 1 / 2 s + 1 + 1 2 s + 1 s 2 + 1 = 1 2 1 s + 1 + 1 s 2 + 1 + s s 2 + 1 y ( t ) = 1 2 ( e - t + cos t + sin t ) b) y 00 + y = cos(2 t ), for t > 0 , y (0) = 1 , y 0 (0) = 0 .

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s 2 ˆ y ( s ) - sy (0) - y 0 (0) + ˆ y ( s ) = s s 2 + 4 ˆ y ( s ) = s ( s 2 + 1)( s 2 + 4) + s s 2 + 1 = s 3 1 s 2 + 1 - 1 s 2 + 4 + s s 2 + 1 = 4 3 s s 2 + 1 - 1 3 s s 2 + 4 y ( t ) = 4 3 cos t - 1 3 cos 2 t 4 (8). An electric circuit gives rise to the system
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Sol-HW6 - Math Methods Fall 2010 Solutions for Assignment 6...

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