Question

Find the inverse Laplace transform of [s^2+1/(s+1)(s-2)^2]?​

Answer 1

Answer:

rs

order by

votes

Up vote

7

Down vote

Accepted

I am going to evaluate this using residues. If you have no idea of what these are, then I will just give you an easy-to-understand intermediate result: if f(s)=p(s)/q(s) and q has a zero at s0, then the residue of f at the pole s=s0 is

p(s0)q′(s0)

The inverse LT of the given

f^(s)=s+1(s2+1)(s2+4s+13)

is simply the sum of the residues of f^(s)est at the poles. The poles here are at s1=i, s2=−i, s3=−2+i3, and s4=−2−i3. I will allow you to apply the above formula to compute the residues of f^(s)est and add them up. I get for the final result,

f(t)=−e−2t[115sin3t+120cos3t]+120(cost+2sint)