Solve the IVP with Laplace Transform: begin{cases} y"+4y'+4y=(3+t)e^{-2t} y(0)=2 y'(0)=5 end{cases}

tinfoQ 2020-12-21 Answered
Solve the IVP with Laplace Transform:
{y"+4y+4y=(3+t)e2ty(0)=2y(0)=5
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Expert Answer

Sadie Eaton
Answered 2020-12-22 Author has 104 answers
Step 1
The equation is:
y"+4y+4y=(3+t)e2t
Now Laplace equation will be:
[s2L{y}sy(0)y(0)]+4[sL{y}y(0)]+4L{y}=31s+2+1(s+2)2
(s2+4s+4)L{y}2s58=3s+2+1(s+2)2
(s2+4s+4)L{y}2s13=3s+2+1(s+2)2
(s2+4s+4)L{y}=2s+13+3s+2+1(s+2)2
Step 2
L{y}=2s3+21s2+63s+59(s+2)2(s)+4s+4
L{y}=2s3+21s2+63s+59(s+2)4
Now take Laplace inverse,
y=L1{2s3+21s2+63s+59(s+2)4}
By the formula of Laplace transformation the final solution will be:
y=(2+9x)e2x
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