Use properties of the Laplace transform to answer the following (a) If f(t)=(t+5)^2+t^2e^{5t}, find the Laplace transform,L[f(t)] = F(s). (b) If f(t)

fortdefruitI 2021-02-25 Answered
Use properties of the Laplace transform to answer the following
(a) If f(t)=(t+5)2+t2e5t, find the Laplace transform,L[f(t)]=F(s).
(b) If f(t)=2etcos(3t+π4), find the Laplace transform, L[f(t)]=F(s). HINT:
cos(α+β)=cos(α)cos(β)sin(α)sin(β)
(c) If F(s)=7s237s+64s(s28s+16) find the inverse Laplace transform, L1|F(s)|=f(t)
(d) If F(s)=e7s(1s+ss2+1) , find the inverse Laplace transform, L1[F(s)]=f(t)
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Expert Answer

Aamina Herring
Answered 2021-02-26 Author has 85 answers
Step 1
We use the results from table of Laplace transform.
a)f(t)=(t+5)2+t2e5t=t2+10t+25+t2e5t
L{f(t)}=L{t2+10t+25+t2e5t}=L{t2}+10L{t}+L{25}+L{t2e5t}
F(s)=2s3+10s2+25s+2(s5)3
b)f(t)=2etcos(3t+π4)2et[cos(3t)cos(π4)sin(3t)sin(π4)]
2etcos(3t)cos(π4)2etsin(3t)sin(π4)
L{f(t)}=2cos(π4)L{etcos(3t)}2sin(π4)L{etsin(t)}
we take constant term Laplace
F(s)=2s(s+1)2+92(s+1)2+9
c)F(7s237s+64s(s28s+16))=4s+3(s4)+7(s4)2
(by partial fraction)
Then L1{F(s)}=L1{4s}+L1{3(s4)}+L1{7(s4)2}
f(t)=4+3e4t+7te4t
d)F(s)=e7s(1s+s(s2+1))=e7ss+se7s(s2+7)
L1{F(s)}=L1{e7ss}+L1{se7s(s2+7)}
=u(t7)+u(t7)cos(t7)
All the results teple of Laplace transform
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