I am trying to understand the method of finding the general solution to the 2nd order hyperbolic partial differential equation
$$v_{\xi\eta} - v = 0$$
described in this answer.
I am able to derive the equation
$$p\hat{v}_{\xi}-\hat{v}=v_{\xi}(\xi,0)$$
where $\hat{v}(\xi,p)$ is the Laplace transform of $v(\xi,\eta)$ taken on the 2nd variable.
However, I get stuck at the following step:
integrate the equation $p\hat{v}_{\xi}-\hat{v}=v_{\xi}(\xi,0) \overset{\rm not}{=}f(\xi)$ to find that $$ \hat{v}(\xi,p)=\int\limits_0^{\xi}f(s) \frac{e^{\frac{\xi-s}{p}}}{p}\,ds+ c(p)e^{\frac{\xi}{p}}. $$
I have puzzled over it quite a bit, and searched for explanations on google, but so far without making progress. Integrate the equation how? What are one or more of the intermediate steps?