Solving a first order nonlinear ODE (nonseparable)

Question

$$ f(x)=\int_{0}^{x}\frac{1}{1-af(t)}dt $$ How would one go about solving this equation? Does this equation have an analytical solution? I have only learned different methods for solving linear ODE and PDEs, so I'm stumped by this equation. All I was able to do was get it into this form: $$ f'(x)=\frac{1}{1-af(x)}, f(0) = c $$ $$ f'(x)-af'(x)f(x)=1 $$ Which did not help me at all in finding a solution. As far as I can tell, this equation is not separable. Suggestion will be great on methods for solving such equations.

Answer 1

set $y=f(x)$, then we have $$\frac{dy}{dx}=\frac{1}{1-a y}$$ or $$(1-a y)dy=dx$$ You can then solve $x$ as a function of $y$.

Answer 2

Hint: It looks like the left side of your last equation can be written as $$\left(f(x) - \frac{a}{2}f(x)^2\right)' $$