Solve y′(t)=sin(t)+int_0^t y(x)cos(t−x)dx such that y(0)=0

figoveck38 2022-11-21 Answered
Solve y ( t ) = sin ( t ) + 0 t y ( x ) cos ( t x ) d x by Laplace transform
My try:
I applied Laplace transform on both sides of the equation.
s L { y ( t ) } = 1 s 2 + 1 + L { c o s ( t ) y ( t ) } s L { y ( t ) } = 1 s 2 + 1 + L { c o s ( t ) } × L { y ( t ) }
Now, I'm stuck on applying the inverse Laplace transform on (*) to find y ( t )
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Answers (2)

Zackary Hatfield
Answered 2022-11-22 Author has 14 answers
Hint. You are on the right track. But please check your results, since from your identity
s L { y ( t ) } ( s ) = 1 s 2 + 1 + L { c o s ( t ) } ( s ) × L { y ( t ) } ( s )
using
L { c o s ( t ) } ( s ) = s s 2 + 1
I rather get
L { y ( t ) } ( s ) = 1 s 3
which is now standard to solve.
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evitagimm9h
Answered 2022-11-23 Author has 5 answers
s L { y ( t ) } = 1 s 2 + 1 + L { c o s ( t ) y ( t ) }
s L { y ( t ) } = 1 s 2 + 1 + L { y ( t ) } s s 2 + 1
L { y ( t ) } ( s s s 2 + 1 ) = 1 s 2 + 1
Here you made a mistake I guess
L { y ( t ) } ( s 3 s + s s 2 + 1 ) = 1 s 2 + 1
L { y ( t ) } = 1 s 3
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