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me375_sp11_hwk03_soln

me375_sp11_hwk03_soln - ME 375 Spring 2011 Homework No 3...

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ME 375 – Spring 2011 Homework No. 3 Due: Friday, February 4 Problem No. 1 (30%) a) For u t ( ) shown below, determine the Laplace transform L u t ( ) . b) If Y s ( ) = s + 6 5 s 2 + 4 s is the Laplace transform of y t ( ) : i. what is y t = 0 ( ) ? ii. what is y t → ∞ ( ) ? c) If Y s ( ) = s + 6 5 s 2 + 4 s is the Laplace transform of y t ( ) , what is the Laplace transform of e 2 t y t ( ) ? Part a): For the function u t ( ) : u t ( ) = 2 h t 4 ( ) + 0.5 t 4 ( ) h t 4 ( ) where h t ( ) is the unit step function. Taking the Laplace transform of u t ( ) above gives: L u t ( ) = 2 L h t 4 ( ) + 0.5 L t 4 ( ) h t 4 ( ) = 2 e 4 s L h t ( ) + 0.5 e 4 s L t [ ] = 2 e 4 s s + 0.5 e 4 s s 2 = 2 s + 0.5 s 2 e 4 s Part bi): y t = 0 ( ) = lim s →∞ sY s ( ) = lim s →∞ s s + 6 ( ) 5 s 2 + 4 s = lim s →∞ s + 6 5 s + 4 = 1 5 Part bii): y t → ∞ ( ) = lim s 0 sY s ( ) = lim s 0 s s + 6 ( ) 5 s 2 + 4 s = lim s 0 s + 6 5 s + 4 = 6 4 = 1.5 Part c): L e 2 t y t ( ) = Y s + 2 ( ) = s + 2 ( ) + 6 5 s + 2 ( ) 2 + 4 s + 2 ( ) = s + 8 5 s 2 + 24 s + 28 2 2
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Problem No. 2 (40%) a) Determine the solution x t ( ) for the following differential equation with the initial conditions of x 0 ( ) = 2 : 10 x + 5 x = 20 h t ( ) , where h t ( ) is the unit step function. b) Determine the solution x t ( ) for the following differential equation with the initial conditions of x 0 ( ) = 0 and x 0 ( ) = 2 :  x + 12 x + 100 x = 0 c)
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