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

tinfoQ

tinfoQ

Answered question

2020-12-21

Solve the IVP with Laplace Transform:
{y"+4y+4y=(3+t)e2ty(0)=2y(0)=5

Answer & Explanation

Sadie Eaton

Sadie Eaton

Skilled2020-12-22Added 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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