Using Laplace Transforms, find the solution to y′(t)+y(t−1)=t^2, with y(t)=0 for t <= 0.

steveo963200054 2022-09-13 Answered
Using Laplace Transforms, find the solution to y ( t ) + y ( t 1 ) = t 2 , with y(t)=0 for t 0
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Answers (1)

Brooklynn Valencia
Answered 2022-09-14 Author has 18 answers
Performing the Laplace transform of the equation you obtain
s Y ( s ) y ( 0 ) = 0 + e s Y ( s ) = 2 s 3
It follows
Y ( s ) = 2 s 3 1 s + e s = 2 s 4 1 1 + 1 s e s = 2 n = 0 ( 1 ) n s n + 4 e n s
It follows
y ( t ) = η i η + i d s 2 π i e s t Y ( s ) = 2 n = 0 ( 1 ) n η i η + i d s 2 π i e s t 1 s n + 4 e n s
= n = 0 2 ( 1 ) n ( n + 3 ) ! ( t n ) n + 3 Θ ( t n ) .
Where we've used Cauchy theorem in the last line.

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