# 4_6b - Solutions 4.6-Page 322 Problem 14 Verify that u (t...

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Solutions 4.6-Page 322 Problem 14 Verify that u ) ( ) ( a t a t = δ by solving the problem 0 ) 0 ( ; ) ( = = x a t x to obtain ) ( ) ( a t u t x = Using direct substitution, u x a t = ) ( . So we need to show that . ) ( ) ( a t u t x = The Laplace transform of the differential equation is: {} { } {} s e X e sX e x x s a t x as as as = = = = ) 0 ( ) ( L L L Taking the inverse Laplace yields: {} = s e X as 1 1 L L Theorem 1 on pg.301 states that { } ( ) ) ( ) ( 1 a t f a t u s F e as = L . For s e as 1 L , s s F a a 1 ) ( , = = . Therefore () ) ( 1 a t f a t u s e as = ( t f L . ) a corresponding to s s F 1 ) ( = is . Therefore 1 ) ( = a t f ) ( 1 a t u x s e as = = L Since, , u ) ( a t u x = ) ( ) ( a t a t = .

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Problem 15 This problem deals with a mass on a spring (with constant ) that receives an impulse at time t . Show that the initial value problems m k 0 0 mv p = 0 = 0 ) 0 ( , 0 ) 0 ( ; 0 v x x kx x m = = = + and 0 ) 0 ( , 0 ) 0 ( ); ( 0 = = = + x x t p kx x m δ have the same solution.
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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_6b - Solutions 4.6-Page 322 Problem 14 Verify that u (t...

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