IEVS2

Solve $u(x) = x + \int_0^x (x-y)u(y)\,dy$

In this problem $f(x) = x,\,\,\,k(x-y)=x-y\,$

Take the Laplace transform of both sides.

$\mathcal{L}u = \mathcal{L}x + \mathcal{L}x\mathcal{L}u\,$

$\mathcal{L}u = \frac{\mathcal{L}x}{1-\mathcal{L}x}\,$ , where $\mathcal{L}x = \frac{1}{s^2}\,$

The solution is

$u(x) = \mathcal{L}^{-1}\left(\frac{1}{s^2-1}\right) = \frac{1}{2}\left(e^x-e^{-x}\right)\,$