Solution of the following initial value problem using the Laplace transform y"+4y=4t y(0)=1 y'(0)=5

remolatg 2021-01-05 Answered
Solution of the following initial value problem using the Laplace transform
y"+4y=4t
y(0)=1
y(0)=5
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Expert Answer

casincal
Answered 2021-01-06 Author has 82 answers
Step 1
Given initial value problem is y"+4y=4t
y(0)=1
y(0)=5
We find solution by using the Laplace transform
Step 2
Solution:
Given initial value problem is y"+4y=4ty(0)=1y(0)=5
Taking Laplace transform on both side we get
L{y+4y}=L{4t}
L{y}+4L{y}=4L{t}
(s2y(s)sy(0)y(0))+4y(s)=4s
s2y(s)s(1)5+4y(s)=4s2
s2y(s)s5+4y(s)=4s2
=(s2+4)y(s)=4s2+s+5
y(s)=4s2(s2+4)+ss2+4+5s2+4
Now e apply inverse Laplace transform to get solution of the IVP
y(t)y(t)=L1[4s2(s2+4)+s(s2+4)+5(s2+4)]
L1[4s2(s2+4)]+L1[ss2+4]+L1[5s2+4]
L1[1s21s2+4]+L1[ss2+4]+L1[5s2+4]{partial fraction 4s2(s2+4)=1s21s2+4}
L1[1s2]L1[1s2+4]+L1[ss2+4]+5L1[1s2+4]
=t12sin(2t)+cos(2t)+52sin(2t)
=t+42sin(2t)+cos(2t)
y(t)=t+2sin(2t)+cos(2t)
Therefore solution of the IVP y+4y=4t,y(0)=1,y(0)=5 is y(t)=t+2sin(2t)+cos(2t)
Step 3
Answer:
The solution of the IVP

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