I am trying to find a solution for this equation: $$f(t)-kt+\int_0^tg(t-\tau)\frac{df^2}{d\tau}d\tau=0$$ I know the $g(t)$ function and I have initial conditions $f(0)$ and $f'(0)$, but I really don't know if it is possible to find an expression for the $f(t)$ function. I tried using Laplace transform since it is useful for the convolution, but I have obtained: $$\mathscr{L}(f(t))-\frac{k}{s^2}+2\mathscr{L}(g(t))\mathscr{L}(f(t)\cdot \frac{df(t)}{dt})=0.$$ I really don't know how to continue (of course I have the expression for $\mathscr{L}(g(t))$, but it is quite complicated so I do not report it here). I would really appreciate if someone can help me with some suggestions or ideas.
Thank you in advance.