Use Laplace transform to solve the folowing initial value problem y"+2y'+y=4e^{-t} y(0)=2 y'(0)=-1

Lewis Harvey 2021-01-22 Answered
Use Laplace transform to solve the folowing initial value problem y"+2y+y=4ety(0)=2y(0)=1
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yunitsiL
Answered 2021-01-23 Author has 108 answers
Step 1
Given : y"+2y+y=4ety(0)=2y(0)=1
Applying Laplace Transform
L(y")+2L(y)+L(y)=4L(e(t))
s2Y(s)sy(0)y(0)+2[sY(s)y(0)]+Y(s)=4(s+1)
(s2+2s+1)Y(s)2s+14=4(s+1)[y(0)=2,y(0)=1]
(s+1)2Y(s)=4(s+1)+2s+3
Y(s)=4(s+1)3+(2s+3)(s+1)2
Y(s)=4(s+1)3+2(s+32+11)(s+1)2
Y(s)=4(s+1)3+2(s+1)(s+1)2+1(s+1)2
Y(s)=4(s+1)3+2(s+1)(s+1)2+1(s+1)2
Step 2
Now taking inverse Laplace Transform,
L1{Y(s)}=4L1{4(s+1)3}+2L1{2(s+1)(s+1)2}+L1{1(s+1)2}
Using the identity L1{1(s+a)n}=tn1eat(n1)! , we get
y(t)=2t2et+2et+tet
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