How to solve equation of the type \begin{align*} \int_{x}^{x+a} f(u) du=e^{-\lambda x} \int_{x-a}^{x} f(u) du \end{align*} we want to solve for $f(x)$ where $\lambda,a$ are some constants.
Things I tried \begin{align*} \frac{d}{dx}\int_{x}^{x+a} f(u) du= \frac{d}{dx} e^{-\lambda x} \int_{x-a}^{x} f(u) du\\ f(x+a)-f(x)= -\lambda e^{-\lambda x} \int_{x-a}^{x} f(u) du+e^{-\lambda x} (f(x)-f(x-a)) \end{align*}
But how to proceed next? Should, I take derivative one more time? But I don't think this will lead anywhere and plus we would have to assume that $f(x)$ is differential.
@Drone Scientist suggested that we can take derivative with respect to $a$. However, since $a$ is constant I am not sure if it is allowed?
Thank you