The given inverse Laplace transform is:
L−1[5s2+12s−4s3−2s2+4s−8]
First split it up into three separate fractions and factorize the denominator
L−1[5s2(s−2)(s2+4)]+L−1[12s(s−2)(s2+4)]−L−1[4(s−2)(s2+4)]
Now to compile them into partial fraction decomp. form:
L−1[5s2(s−2)(s2+4)]=A(s−2)+Bs+c(s2+4)
L−1[12s(s−2)(s2+4)]=As−2+Bs+cs2+4
L−1[4(s−2)(s2+4)]=As−2+Bs+cs2+4
Then following through:
5s2=A(s2+4)+(Bs+c)(s−2)
12s=A(s2+4)+(Bs+c)(s−2)
4=A(s2+4)+(Bs+c)(s−2)
Now to solve for the variables, first with s=2:
5∗4=8A→A=208
12(2)=8A→A=3
4=8A→A=12
Now subbing in for s=0:
0=208(02+4)+(B(0)+C)(0−2)→0=10−2C→C=5
12(0)=3(4)+(B(0)+C)(−2)→0=12−2C→C=6
4=12(4)−2C→4=2−2C→C=−1
Now to solve for B; s=1
5=1008−B−5→B=52
12=3(1+4)+(B+6)(−1)→B=−3
4=12(1+4)+(B−1)(−1)→B=−12
So now our Laplace transforms are:
L−1[5s2(s−2)(s2+4)]=208(s−2)+5s+52(s2+4)
L−1[12s(s−2)(s2+4)]=3s−2−3s+6s2+4
L−1[4(s−2)(s2+4)]=12(s−2)−s−12(s2+4)
Are my calculations correct or have I over complicated the problem?
