Find the inverse Laplace Tranformation by using convolution theorem for the function frac{1}{s^3(s-5)}

glasskerfu 2021-03-05 Answered
Find the inverse Laplace Tranformation by using convolution theorem for the function 1s3(s5)
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Expert Answer

Bella
Answered 2021-03-06 Author has 81 answers

Step 1
According to the given information, it is required to find the inverse of Laplace transform using convolution theorem.
1s3(s5)
it is required to find L1(1s3(s5))
Step 2
Solving further:
let G(s)=1s3,H(s)=1(s5)
the inverse laplace of G(s) and H(s) are
L1(G(s))=L1(1s3)=t22=g(t)
L1(H(s))=L1(1(s5))=e5t=f(t)
Step 3
The convolution theorem states that if G(s) and H(s) are the Laplace transform of g(t) and h(t) then,
L((gh)(t))=G(s)H(s)
(gt)(t)=L1(G(s)H(s))
g(t)h(t)=L1(F(s))=0tf(tu)g(u)du
Step 4
Now solve the integral to find the inverse.
0te5(tu)u22du=12e5t0t(e5uu2)du
=12e5t[1125(25t2e5t2(5te5te5t)2)]
=1250e5t[25t2e5t+10te5t+2e5t2]
0te5(tu)u22du=L1(1s3(s5))=110t2125t1125+1125e5t

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