Thomas,

I fixed you LaTeX (it was a double subscript error \int_{0{}_{t} but your formula is wrong: you should integrate by $x$ (inner) from $x-c(t-s)$ to $x+c(t-s)$ and integrand is $e^{x-s}$ and then you integrate by $s$ from $0$ to $t$; also forgot $\frac{1}{2c}$ factor.

I have not checked other terms. You can see that your "solution" does not satisfy initial conditions (or equations)