  The autonomous differential equation in the title is derived in
S.Srinivasan~\cite{ss05} (equation (E) in ~\cite{ss05}) in the context of
certain discrete sums from the number theoretic considerations. These discrete
sums are then estimated in terms of an integral involving the solutions of this
differential equation; no analysis is done on this integral in ~\cite{ss05}.
Our main objective is to consider this equation from the dynamical systems view
point and describe all the orbits in the phase plane. The equation is singular
in the sense that the coefficient of $\ddot{x}$ vanishes or becomes unbounded
at a few points. This singularity puts a restriction on the initial data. The
solutions that we obtain in explicit form are, however, smooth and satisfy the
equation pointwise or in the limiting sense. The explicit form of solutions may
also be used to analyse the integral in ~\cite{ss05}. The equation possesses an
infinite number of equilibrium or critical points and there are periodic orbits
surrounding each of them in a part of the phase plane. In the complement of
this part of the phase plane, there are non-periodic orbits, any two of which
have an infinite number of common points. This, however, is in great contrast
with the fact that any two (distinct) orbits of a regular autonomous system are
disjoint, thus bringing out the singular nature of the equation
