Solve the integral equation y(t)=e^t(1+int_0^t e^(-tau) y(tau)d tau) with Laplace transform

Jacoby Erickson 2022-10-13 Answered
Solve the integral equation y ( t ) = e t ( 1 + 0 t e τ y ( τ ) d τ ) with Laplace transform
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Answers (2)

Messiah Trevino
Answered 2022-10-14 Author has 18 answers
Let the Laplace of y(t) be Y(s).
Applying Laplace on both the sides we end up with
> Y ( s ) = 1 s 1 + ( L [ e t ] ) ( L [ y ( t ) ] )
> Y ( s ) = 1 s 1 + 1 s 1 Y ( s )
Solving we end up in,
> Y ( s ) = 1 s 2
Taking inverse Laplace we get, y ( t ) = e 2 t
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raapjeqp
Answered 2022-10-15 Author has 2 answers
y ( t ) = e t ( 1 + 0 t e τ   y ( τ ) d τ )
y ( t ) = e t ( 1 + 0 t e τ   y ( τ ) d τ ) + y ( t ) = 2 y ( t ) y ( t ) = C e 2 t
Substitute into the original equation to find C = 1
C e 2 t = e t ( 1 + C 0 t e τ d τ )
C e t = 1 + C ( e t 1 ) C = 1
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