I am working on something else, but this is a point that I do not know how to do. $g(x) = \int^x_0 -\frac{x}{(a-y)^2} f(y) dy$ given $(Lf)(x) = g(x)$. How should I approach finding the inverse $L^{-1}$ in this case.
I tried differentiate both sides and got: $$g'(x) = -\frac{x}{(a-x)^2}f(x) + \int^x_0 -\frac{1}{(a-y)^2}f(y)dy$$ Then I used $g(x)/x = \int^x_0 -\frac{1}{(a-y)^2}f(y)dy$ and replace it back in to get a differential equation, which led me nowhere.
How should I start solving for $L^{-1}$ or does it even exist?