Solving an integral equation.

Question

Let $K(x,y)=(x+y)e^{-y^2/2-xy}$. I need a constructive way (not simply verifying it is a solution) to show that $f(x)=e^{-x^2/2}$ is the solution to the integral equation: \begin{equation}f(y)=\int_0^\infty K(x,y)f(x)\text{ d}x.\end{equation} Any help is appreciated.