Review Problem 5 solution

Review Problem 5 solution - 5. If u ( t ) = 3 (1 - e -3t )...

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5. If ( ) ( ) 3 3 1 t u t e - = - and ( ) 0 0 x = while ( ) 0 0 x = d , what is the total time solution for ( ) x t ? (Although you could use ( ) ( ) 3 3 3 3 3 0 3 5 8 t t G G e e - - - - = - to find the particular solution, we will use Laplace Transform method to find the total solution.) The Laplace Transform Table gives the transform pairs of 1 , 0 ! n at t e t n - and ( ) 1 1 n s a + + . If 0 n = and 0 a = the pair degenerates to a unit step and 1 s . If 0 n = and 3 a = the pair degenerates to 3 t e - and 1 3 s + . Thus, by superposition, the Laplace Transform of ( ) ( ) 3 3 1 t u t e - = - is ( ) ( ) 1 1 9 3 3 3 U s s s s s = - = + + . From Problem 1, ( ) ( ) ( ) 0 0 2 2 2 2 5 2 5 U s s x x X s s s s s + + = + + + + + d . As the initial conditions are given and the input has been rewritten in frequency space, the desired output is ( ) ( ) ( ) ( ) 2 2 9 1 2 5 3 2 5 U s X s s s s s s s = = + + + + + Partial fraction expansion requires solving the expression ( ) ( ) 2 2 9 1
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Review Problem 5 solution - 5. If u ( t ) = 3 (1 - e -3t )...

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