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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

$$x(t)=\frac{c_{1}}{A^{2}}e^{-At}+c_{2}t+c_{3}$$

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    $\begingroup$ The linear equation $\dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}\left(c_{2}\cos\frac{\sqrt{3}}{2}t+c_{3}\sin\frac{\sqrt{3}}{2}t\right)$$ $\endgroup$ – JennyToy Jan 5 at 23:55

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