I want to find the solution of the following differential equation.
$$ \frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} = −\frac{1}{ax + \mathrm{d}x/\mathrm{d}t} $$
where a is a positive constant and x is a position of a free object that is not bounded or tied to a fixed object (e.g., a wall).
I think that the solution may behave like an oscillator, but it seem to be slightly different from that of the oscillator. However, due to the terms in the denominator, it has the singularity. How to solve this type of equation?
** As I studied differential equations for a while 20 years ago, I do not know how to solve this problem. Since the solution of the above differential equation is an important issue for me, I would appreciate it if you help.