Methods for solving non-linear ode

Question

I had the second order non-linear ODE $$f''\left(t\right)f'\left(t\right)=f\left(t\right)^{2}$$ And I managed to reduce it to a first order ODE $$f'\left(t\right)=\left(C^3+f\left(t\right)^{3}\right)^{1/3}$$ and noticed that I can set $C=1$ and reintroduce it at the end, since it just scales the solution. But I'm stuck at this point. Would appreciate any help, thanks.

*I know that $f\left(t\right)=e^{kt}$ is a solution when $k$ is a cube root of unity, but I'm after the general solution.

Answer 1

Substituting $$v(y)=\frac{dy(x)}{dx}$$ then you will getz $$\frac{dv(y)}{dy}v(y)^2=y^2$$ $$\int\frac{dv(y)}{dy}v(y)^2dy=\int y^2dy$$ $$\frac{v(y)^3}{3}=\frac{y^3}{3}+C_1$$ can you finish?

Answer 2

The first order equation in $f$ is in separated variables. You can write the solution as $$ \int\frac{df}{\left(C^3+f\left(t\right)^{3}\right)^{1/3}}=t+K $$ for a constant $K$. Unfortunately, this integral is not expressible in terms of elementary functions. Mathematica gives it in terms of Apell hypergeometric functions.