4_5b - Problem 21 Find the Laplace transforms of the given...

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Unformatted text preview: Problem 21 Find the Laplace transforms of the given functions. 2 if ) ( ; 2 1 if 2 ) ( ; 1 if ) ( > = ≤ ≤ − = ≤ = t t f t t t f t t t f Rewriting using the step function yields ) ( t f 2 ) ( t t u + [ ] [ ] ) ( ) ( 2 ) ( ( 2 ) ( ) ( 2 ) ( ) ( 2 ) ( ) 2 ( ) 1 ( ) 2 ( ) 1 1 ) ( 2 2 1 1 2 2 1 1 1 t tu t u t u tu t t tu t u t tu t u t tu t t u t u t t t f + − − = + − − + − = − − − − + = − − { } { } { } { } { } { } { } { } { } { } { } { } ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 2 ) ( 2 ) ( 2 ) ( 2 4 4 2 3 3 1 2 2 1 1 1 2 2 1 1 2 2 1 1 t u a t f t u a t f t u a t f t u a t f t t tu t u t u t tu t t tu t u t u t tu t t f − + − − − + − − = + − + − = + − + − = L L L L L L L L L L L L Theorem states that L . For this problem, ( ) { } ) ( ) ( s F e a t f a t u as − = − − ) 1 ( 2 ) ( 2 ) ( , 1 1 1 1 1 + = ∴ = − = t t f t a t f a 2 ) ( 2 ) ( , 1 2 2 2 2 = ∴ = − = t f a t f a 2 ) ( 2 ) ( , 2 3 3 3 3 = ∴ = − = t f a t f a 2 ) ( ) ( , 2 4 4 4 1 + = ∴ = − = t t f t a t f a Using Figure 4.1.2 to find the Laplace transforms, { } { } { } { } { } s e e e e s e e e s s e s e s e s s s t u a t f t u a t f t u a t f t u a t f t s s s s s s s s s s 2 2 2 2 2 2 2 2 2 2 4 4 2 3 3 1 2 2 1 1 1 2 2 2 2 2 1 2 1 2 2 2 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( − − − − − − − − − − + − + − + + − = + + − + + − = − + − − − + − − L L L L L 2 2 2 1 ) ( s e e s F s s − − + − = Problem 31 Solve the initial value problem ) ( ) ( ; ) ( = ′ = = + ′ + ′ ′ x x t f kx x c x m with the given data π π ≥ = < ≤ = = = = t t f t t f c k m if ) ( , if 1 ) ( ; , 4 , 1 Substituting the given information and using step functions to express the forcing function yields: ) ( 1 4 t u x x π − = + ′ ′ The necessary Laplace transforms needed to transform the equation are given below. { } { } { } { } { } ) ( 1 ) ( 1 ) ( ) ( ) ( ) ( ) ( 2 2 t u t u s X s x sx t x s t x π π L L L L L − = − = ′ − − = ′ ′ Theorem states that L . For this problem, ( ) { } ) ( ) ( s F e a t f a t u as − = − − 1 ) ( , = − = a t f a π ....
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This note was uploaded on 06/06/2011 for the course EGM 3311 taught by Professor Haftka during the Spring '11 term at University of Florida.

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4_5b - Problem 21 Find the Laplace transforms of the given...

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