use the Laplace transform to solve the given initial-value problem y''+5y'+4y=0 y(0)=1 y'(0)=0

opatovaL 2021-01-08 Answered
use the Laplace transform to solve the given initial-value problem
y+5y+4y=0
y(0)=1
y(0)=0
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Expert Answer

falhiblesw
Answered 2021-01-09 Author has 97 answers
Step 1
We know thatL[y]=s2y(s)sy(0)y(0)
L[y]=sy(s)y(0)
Step 2
Given that
y+5y+4y=0
taking Laplace transform both sides,
L[y]+5L[y]+4L[y]=0
(s2y(s)sy(0)y(0))+5(sy(s)y(0))+4y(s)=0
Now use the given initial conditions y(0)=1 and y'(0)=0, so
(s2y(s)s)+5(sy(s)1)+4y(s)=0
s2y(s)s+5sy(s)5+4y(s)=0
y(s)(s2+5s+4)=s+5
y(s)=(s+5)(s2+5s+4)
y(s)=(s+5)(s2+4s+s+4)
y(s)=(s+5)s(s+4)+1(s+4)
y(s)=(s+5)(s+1)(s+4)
Step 3
Now simplify (s+5)(s+1)(s+4) using partial fraction method,(s+5)(s+1)(s+4)=A(s+1)+B(s+4)
s+5=A(s+4)+B(s+1)
Put s=1,4=3AA=43
Put s=4,1=3BB=13
So
(s+5)(s+1)(s+4)=(43)(s+1)(13)(s+4)
Step 4
So
y(s)=(43)(s+1)(13)(s+4)
Now taking inverse Laplace transform,
y=L1[43(s+1)]L1[13(s+4)]
y=43L1[1(s+1)]13L1[1(s+4)]
y=43et13e4t[L1(1(sa))=eat]
Step 5
Hence using the Laplace transform solution of given initial value problem is
y=43et13e4t
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