Solve $$y(x)=e^x(1+\int_{0}^{x}e^{-t}y(t) dt)$$
Here's what I did: take derivative on both sides, $y'(x)=e^x+e^x \int_{0}^{x}e^{-t}y(t)dt +e^xe^{-x}y(x) \Rightarrow y'(x)=e^x+y(x)-e^x+y(x) \Rightarrow y'=2y$. Hence the solution looks like $y(x)=e^{2x}+C$. But I wonder how to solve this equation using Laplace transform and convolution. Any idea?