How to show $|f''|=|1+f'^2|(tf'-f)$ has unique solution?

Question

I calculate the 1-dimensional self-similar solution of mean curvature flow, and found this quesiton is equal to prove $$ |f''|=|1+f'^2|(tf'-f) $$ has unique solution. Where $f=f(t): \mathbb R\rightarrow \mathbb R$, and $f\ge 0$. Obviously, $f\equiv 0$ is a solution of above ODE. But, how to show this ODE has unique solution ?