I am currently studying the following second order ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\ y(0) = 0\\ \dot y(T) = c \end{cases} for $c$ a positive constant and $T > 0$ and arbitrarily small constant. Clearly, solving this ODE explicitly is not doable. But actually for my purpose I don't need a whole solution but rather to understand the behaviour of $\dot y(x)$ near $0$. I would like to know if $\lim_{x \to 0} \dot y(x) = 0 $ or not.

Being not really familiar with the tools used in ODE theory, I was wondering if one of you guide me on this topic. I solved the ODE numerically using *Maple* and the software gave me the following plots

where I fixed $T = 1/2$ and the initial conditions set at $c = 1, 1/2$ and $1/3$ corresponding to $1, 2$ and $3$ respectively. It seems that the slope at the origin is always strictly positive, whatever initial condition I set (if I decrease $c$ or $T$, the behaviour remains the same). However, I have absolutely no idea how to do that. Any help?

I initially asked this question Stackexchange Mathematics but as I haven't got any answer I crossposted it.