hw05solns - 1 Homework 5 Solution Problem 1 Given x + 16x =...

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1 Homework 5 Solution Problem 1 Given ¨ x + 16 x = 0 with x (0) = 1 , ˙ x (0) = 0. Using Laplace transformation, 0= s 2 X ( s ) - sx (0) - ˙ x (0) + 16 X ( s ) X ( s )= s s 2 + 16 x ( t ) = cos 4 t. Problem 2 Given ¨ x + 16 x = 0 with x (0) = 1 , ˙ x (0) = 1. Using Laplace transformation, s 2 X ( s ) - sx (0) - ˙ x (0) + 16 X ( s ) X ( s s s 2 + 16 + 1 s 2 + 16 x ( t ) = cos 4 t + 1 4 sin 4 t. Problem 3 Given ¨ x + 5 ˙ x +6 x = 0 with x (0) = 2 , ˙ x (0) = - 5. Using Laplace transformation, s 2 X ( s ) - 2 s + 5 + 5( sX ( s ) - 2) + 6 X ( s ) X ( s 2 s +5 s 2 s = 1 s +2 + 1 s +3 x ( t e - 2 t + e - 3 t . Problem 4 Given ¨ x = 0 with x (0) = 0 , ˙ x (0) = 1. Using Laplace transformation, s 2 X ( s ) - sx (0) - ˙ x (0) X ( s 1 s 2 x ( t t 1( t ) . Problem 5 Given ¨ x + 2 ˙ x x = 6 cos3 t - 4 sin 3 t with x (0) = 0 , ˙ x (0) = 5. Using Laplace transfor- mation, ( s 2 X ( s ) - 5) + 2 sX ( s ) + 5 X ( s 6 s - 12 s 2 +9 X ( s 6 s - 12 ( s 2 + 9)( s 2 s + 5) + 5 s 2 s = 3 s 2 + 2 s 2 s x ( t ) = sin 3 t + e - t sin 2 t. Problem 6 Given ¨ x + 16 x = δ ( t ) with x (0) = ˙ x (0) = 0. Using Laplace transformation, 1= s 2 X ( s ) - sx (0) - ˙ x (0) + 16 X ( s ) X ( s 1 s 2 + 16 x ( t 1 4 sin 4 t.
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2 Problem 7 Given ¨ x + 16 x = δ ( t - 2) with x (0) = ˙ x (0) = 0. Using Laplace transformation, e - 2 s = s 2 X ( s ) - sx
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hw05solns - 1 Homework 5 Solution Problem 1 Given x + 16x =...

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