Solve the initial value problem x^{(4)}-5x"+4x=1-u_x(t), x(0)=x'(0)=x"(0)=x'''(0)=0

Tazmin Horton 2021-01-10 Answered
Solve the initial value problem
x(4)5x"+4x=1ux(t),
x(0)=x(0)=x"(0)=x(0)=0
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Expert Answer

nitruraviX
Answered 2021-01-11 Author has 101 answers
Step 1
To solve the given initial value problem we use the Laplace transform .Here upi(t) is Heaviside function .
We require few results of Laplace transform and inverse Laplace transform.
L{1}=1s
L{x(n)}=snX(s)k=0n1sn1kxk(0))
L{uc(t)}=ecss
L{1sn+1}=tn(n!)
L{ecsF(s)}=uc(t)f(tc)
Step 2
Given initial value problem
x(4)5x"+4x=1uπ(t),x(0)=x(0)=x"(0)=x(0)=0
Applying Laplace transform on both sides of I.V.P. , and using the results in step 1, we get
L{x(4)5x"+4x}=L{1uπ(t)}
L{x(4)}L{5x"}+L{4x}=L{1}L{uπ(t)}
s4X(s)s3x(0)s2x(0)sx"(0)x(0)5(s2X(s)sx(0)x(0))+4X(s)=1seπss
(s45s2+4)X(s)=1seπss
X(s)=1eπss(s45s2+4)
X(s)=1eπss(s+1)(s1)(s+2)(s2)
Applying Inverse Laplace transform to get x(t)
L1{X(s)}=L1(1e(pis))/(s(s+1)(s1)(s+2)(s2))
x(t)=e2t4et+64et+e2t24uπ(t)e2(tπ)4etπ+64e(tπ)+e2(tπ)24
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