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

Solve the integral equation $y\left(t\right)={e}^{t}\left(1+{\int }_{0}^{t}{e}^{-\tau }y\left(\tau \right)d\tau \right)$ with Laplace transform
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Messiah Trevino
Let the Laplace of y(t) be Y(s).
Applying Laplace on both the sides we end up with
$\begin{array}{rl}>Y\left(s\right)& =\frac{1}{s-1}+\left(L\left[{e}^{t}\right]\right)\left(L\left[y\left(t\right)\right]\right)\end{array}$
$\begin{array}{rl}>Y\left(s\right)& =\frac{1}{s-1}+\frac{1}{s-1}Y\left(s\right)\end{array}$
Solving we end up in,
$\begin{array}{rl}>Y\left(s\right)& =\frac{1}{s-2}\end{array}$
Taking inverse Laplace we get, $y\left(t\right)={e}^{2t}$
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raapjeqp

Substitute into the original equation to find $C=1$
$C{e}^{2t}={e}^{t}\left(1+C{\int }_{0}^{t}{e}^{\tau }d\tau \right)$
$C{e}^{t}=1+C\left({e}^{t}-1\right)\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}C=1$
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