Let x(t) be the solution of the initial-value problem(a) Find the Laplace transform F(s) of the forcing f(t).(b) Find the Laplace transform X(s) of the solution x(t).x"+8x'+20x=f(t)x(0)=-3x'(0)

necessaryh 2020-12-25 Answered

Let x(t) be the solution of the initial-value problem
(a) Find the Laplace transform F(s) of the forcing f(t).
(b) Find the Laplace transform X(s) of the solution x(t).
x"+8x+20x=f(t)
x(0)=3
x(0)=5
where the forcing f(t) is given by 
f(t)={t2for 0t<2,4e2tfor 2t<.

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Expert Answer

lamusesamuset
Answered 2020-12-26 Author has 93 answers

given differential equation is x
x"+8x+20x=f(t)
x(0)=3
x(0)=5
where the forcing f(t) is given by 
f(t)={t2for 0t<2,4e2tfor 2t<.
 x"+8x+20x=t2+4e2t
we know that L[x"]=s2X(s)sX(0)x(0)
L[x]=sX(s)x(0)
L[tn]=n!sn+1,L{eat}=1s+a
Step 2
Taking Laplace tranform of equation
L[x"]+8L[x]+20L[x]=L[t2]+4e2L{et}
s2(X(s))sX(0)x(0)+8[s(X(s))x(0)]+20X(s)=2!s2+1+4e2s+1
s2X(s)s(3)5+8(sX(s)5)+20X(s)=2s3+4e2s+1
(s2+8s+20)X(s)+3s540=2s3+4e2s+1
(s2+8s+20)X(s)+3s45=2s3+4e2s+1
X(s)[s2+8s+20]=2s3+4e2s+1+453s
x(s)=1s2+8s+20{(453s)+2s3+4e2s+1}

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