2nd order homogeneous ODE I am trying to solve a system of differential equations (for the full system see below) and I am stuck the following 2nd order ODE (with $a$ and $b$ being constants and $\dot x = \frac{dx}{dt}$):
$$\ddot x - \frac34 a\dot x^2 -b \dot x + 2 a b = 0$$
I tried to substitute $v := \dot x$, which leads me to a different ODE which I was not able to solve. And now I have no idea what else I could try here and would love to get some pointers.
And here the original system / IVP I want to solve:
$$ \dot x = 2 a + 2 e^{ax}y \\ \dot y = \frac32 e^{-ax}a^3+(a^2+b)y-\frac12 a e^{ax}y^2 \\ x(0)=0 \\ y(T)=0$$
In which I solved the first equation for $y$ and substituted $y$ and $\dot y$ in the 2nd which gave me the equation above.

Hint
you can use $\ddot x=\frac{\dot x d\dot x}{dx}$ and then use separation of variables
