Consider the system of 3 ordinary differential equations
$$\dot{x}=v$$
$$\dot{v}=a$$
$$\dot{a}=-Aa+v^{2}-x$$
which can also be written as a single 3rd order ODE
$$\dddot{x}=-A\ddot{x}+\dot{x}^{2}-x$$
$A$ is an arbitrary constant and the dot means derivative with respect to time, i.e. $\dot{x}=dx/dt,\ddot{x}=d^{2}x/dt^{2}$, etc. This system can be thought as describing the time evolution of the position $x$, velocity $v$ and acceleration $a$ of a particle.
Are there any limits where we can solve analytically this system, i.e. find $x(t),v(t),a(t)$?
For example when $A=0$? A perturbative solution would also be good. Or maybe there is a way of reparametrizing time to make the system a known integrable one?
I know that the simpler system
$$\dddot{x}=-A\ddot{x}\iff \dot{a}=-Aa$$
has the solution
$$a(t)=c_{1}e^{-At}$$ which means that
$$x(t)=\frac{c_{1}}{A^{2}}e^{-At}+c_{2}t+c_{3}$$