# Analytic or perturbative solution in any limits?

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

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

• The linear equation $\dddot{x}+x=0$ has the general solution $$x(t)=c_{1}e^{-t}+e^{t/2}\left(c_{2}\cos\frac{\sqrt{3}}{2}t+c_{3}\sin\frac{\sqrt{3}}{2}t\right)$$ – JennyToy Jan 5 '19 at 23:55