Determine L^{-1}left[frac{(s-4)e^{-3s}}{s^2-4s+5}right]

FobelloE 2021-03-18 Answered
Determine L1[(s4)e3ss24s+5]
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Expert Answer

yunitsiL
Answered 2021-03-19 Author has 108 answers
Concept used:
The inverse laplace transform rule.
If L1{F(s)}=f(t) then
1. L1{easF(s)}=H(ta)f(ta), Here, H(t) is Heaviside step function.
2.L1{F(sa)}=eatf(t)
3.L1{sF(s)}=f(t)+f(0)
The linearity property of the inverse laplace transform is
L1{af(s)+bg(s)}=aL1{f(s)}+bL1{g(s)}
From the Inverse Laplace transform table.
L1{as2+a2}=sin(at)
Calculation
Apply the inverse transform rule "If L1{F(s)}=f(t)
then L1{easF(s)}=H(ta)f(ta)".
For the function (s4)e3ss24s+5,
F(s)=s4s24s+5,a=3
L1{e3ss4s24s+5}=H(t3)L1{F(s3)}(1)
First find L1{F(s)}=f(t) where F(s)=s4s24s+5
L1{s4s24s+5}=L1{s2(s2)2+121(s2)2+1}
Step 3
Now apply the linearity property of inverse Laplace transform.
L1{s4s24s+5}=L1{s2(s2)2+121(s2)2+1}
L1{s2(s2)2+1}2L1{1(s2)2+1}
 Now apply the inverse transform rule L1{F(sa)}=eatL1(F(s))
L1{s2(s2)2+1}2L1{1(s2)2+1}=e2tL1{ss2+1}2e2tL1{1s2+1}(2)
Step 4
Now solve L1{1s2+1} by the use of
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