How to solve for the equation y'''+4y''+5y'+2y = 4x+16 using laplace transform method given that y(0) = 0, y'(0) = 0, text{and } y''(0) = 0

Tahmid Knox 2020-10-31 Answered
How to solve for the equation
y+4y+5y+2y=4x+16
using laplace transform method given that
y(0)=0,y(0)=0,and y(0)=0
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Expert Answer

okomgcae
Answered 2020-11-01 Author has 93 answers
Step 1
We use identities of Laplace transform then we take Inverse Laplace Transform.
Step 2
Given ODE is,
y+4y+5y+2y=4x+16
We know that L[y]=s3y(s)s2y(0)sy(0)y"(0)
L[y"]=s2y(s)sy(0)y(0)
L[y]=sy(s)y(0),L[y]=y(s)
L[x]=1s2,L[1]=1s
Given conditions all y(0)=0,y(0)=0,y"(0)=0
Taking Laplace tranform of y+4y+5y+2y=4x+16 from
L[y]+4L[y"]+5L[y]+2L[y]=4L[x]+16L[1]
s3y(s)s2y(0)sy(0)y"(0)+4[s2y(s)sy(0)y(0)]+5[s(y(s))y(0)]+2y(s)=4s2+16s
(s3+4s2+5s+2)y(s)=4(1+4ss2)
(s+1)(s+1)(s+2)y(s)=4[4s+1s2]
y(s)=4[4s+1]s2(s+1)2(s+2)
Taking inverse Laplace transformy(t)=4L1{4s+1s2(s+1)2(s+2)}
=3H(t)+2t+4etet12t7e2t
y(t)=3H(t)+2t+(412t)et7e2t
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