This is from Berkeley Problems in Mathematics, Spring 86. It asks for $\lambda\in \mathbb{R}$, find all solutions of the following two equations:
$\phi(x)=e^{x}+\lambda\int^{x}_{0}e^{x-y}\phi(y)dy; \psi(x)=e^{x}+\lambda\int^{1}_{0}e^{x-y}\psi(y)dy$
My thought is to take the derivative, thus we have $\frac{d}{dx}\phi=e^{x}+\lambda\phi(x)$ because we have $\frac{d}{dx}\lambda \int^{x}_{0}e^{x}/e^{y}\phi(y)dy=\lambda e^{x}/e^{x}\phi(x)=\lambda \phi(x)$. And so is the equation for $\psi$ Thus the difference equation would be $\frac{d}{dx}(\phi-\psi)=\lambda(\phi-\psi)$ and implies $\phi-\psi=Ce^{\lambda x}$ for some $C$. But I do not know how to use this to solve the original equation. The seeming simple equation $\frac{d}{dx}\phi=e^{x}+\lambda\phi(x)$is also not easy to solve. I do not know how to treat the delay term $e^{x}$ or find a special solution for this.