Solve the following IVP using Laplace Transformy′′+3y′+2y=e^(-t), y(0)=0 y′(0)=0

Tobias Ali 2021-03-09 Answered

Solve the following IVP using Laplace Transform
y+3y+2y=et,y(0)=0y(0)=0

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Expert Answer

faldduE
Answered 2021-03-10 Author has 109 answers

Step 1
We have to solve given IVP using Laplace transform.
Step 2
We have given
y+3y+2y=et,y(0)=0y(0)=0
Taking Laplace transform on given differential equation
L(y+3y+2y)=L(et)
L(y)+3L(y)+2L(y)=1s+1(i)
We are using here
L(y)=s2Y(s)sy(0)y(0),L(y)=sY(s)y(0)  and  L(et)=1s+1
where L(y)=Y(s)
Then from equation (i) we get,
s2Y(s)sy(0)y(0)+3(sY(s)y(0))+2Y(s)=1s+1
s2Y(s)+3sY(s)+2Y(s)=1s+1
(s2+3s+2)Y(s)=1s+1
(s(s+1)+2(s+1))Y(s)=1s+1
(s+1)(s+2)=1s+1
Y(s)=1(s+1)2(s+2)(ii)
Now
1(s+1)2(s+2)=As+1+B(s+1)2+Cs+2
1(s+1)2(s+2)=A(s+1)(s+2)+B(s+2)+C(s+1)2(s+1)2(s+2)
1=A
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