Use Laplace to solve IVP: y′′−2y′3y=3e^t;y(0)=−1,y′(0)=2

Simone Watts

Simone Watts

Answered question

2022-11-18

Use Laplace to solve IVP:
y 2 y 3 y = 3 e t ; y ( 0 ) = 1 , y ( 0 ) = 2

Answer & Explanation

embutiridsl

embutiridsl

Beginner2022-11-19Added 26 answers

Hints:
Solve for y(s).
Combine all terms and then write out using Partial FractionsFind the Inverse Laplace Transform of the previous steps
We have:
L [ 3 e t ] = 3 s 1
L [ y ] = s 2 y ( s ) s y ( 0 ) y ( 0 ) = s 2 y ( s ) + s 2
L [ 2 y ] = 2 ( s Y ( s ) y ( 0 ) ) = 2 s y ( s ) 2
L [ 3 y ] = 3 y ( s )
We now find y(s) as:
( s 2 2 s 3 ) y ( s ) = 3 s 1 s + 4
So:
y ( s ) = 3 4 ( s 1 ) 7 8 ( s + 1 ) + 5 8 ( s 3 )
Now just find the inverse of each of those.
Final result:
y ( t ) = 1 8 e t ( 6 e 2 t + 9 e 4 t + 5 )

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