I'm studying how to solve differential equations through superposition of Green's function and so far everything makes sense in the process of finding the solution in the integral form, but then I'm unable to reverse the steps to check my solution. How can a I differentiate a function in such an integral formula to check whether my solution is correct? Naively, I first assumed I just had to use the FTC, but then I realised the variable of the function involved is not in the limits of integration but inside the integral.
For example, for the simple problem $-cu''(x)=f(x)$, $u(0)=0$, $u'(1)=0$, I get the following solution: $$u(x)=\int_0^1\frac{f(\xi)(x-\rho(x-\xi))}{c}d\xi$$ ($\rho$ stands for the ramp function). How can I derivate such a function with respect to $x$?
My approach would be to split up the integral into two parts: $\int_0^x \dots d\xi + \int_x^1 \dots d\xi$. You can write what's on the dots explicitly. For example, for $\xi \in (0,x),$ the quantity $x - \xi > 0$ so $\rho(x - \xi) = x - \xi.$ After substituting the explicit expressions for $\rho$ you can use this version of Leibniz' rule to compute the derivatives: https://en.wikipedia.org/wiki/Leibniz_integral_rule