Solving $yy''+ky'^2-k=0$

Question

During some calculations about geodesics, after some simplification I found the following Cauchy problem:

$$yy''+ky'^2-k=0$$ $$y(0)=1,y'(0)=c$$

with $c>1$ and $0<k<1$.

How could I solve it?

If $k=1$ then I could use $$(yy')'=y'^2+yy''$$ is there any similar substitution in the case $k\neq1$?

Answer 1

Transform into $yy''=k(1-y'^2)$ and then into the separated form $$ \frac{y'y''}{1-y'^2}=k\frac{y'}{y} $$