I work with a problem that gives me a nonlinear 3rd order ODE. The equation in question is:
$f'''(x) - \alpha f(x) f''(x) + \alpha f'(x)^2 + k f'(x)=0$
The coefficient $k$ always turns out to be an integer, and $\alpha$ is real and positive. I attempted transforming the equation to one of the Weierstrass (elliptical) differential equation forms, but without any success. Of course, there is a trivial solution $f(x) = -\frac{k}{\alpha}x + C_0$. Is there a method to find other analytical solutions (if any, i.e.)?