Integral equations that can be solved elementary 
Solve the following integral equations:
  $$
\int_0^xu(y)\, dy=\frac{1}{3}xu(x) \tag 1 \label 1
$$
  and
  $$
\int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1. \tag 2 \label 2
$$

Concerning $\eqref 1$, I read that it can be solved by differentiation. Differentiation on both sides gives
$$
u(x)=\frac{1}{3}(u(x)+xu'(x))\Leftrightarrow u'(x)=\frac{2}{x}u(x)
$$
and then by separation of this ODE the solution is $u(x)=Cx^2$.
How can I solve $\eqref 2$?

Hint:
$$\int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1$$ So $$  e^{-x}\int_0^xu(y)\, dy=e^{-x}+x-1$$ and therefore, $$ \ \int_0^xu(y)\, dy=1+\frac{x-1}{e^{-x}} $$
