Renormalisation group theory applied to $\ddot{x}+\dot{x}+x^2=0$

The titular ordinary differential equation (ODE) is encountered in the theory of on-axis inertial particle capture by a blunt stationary collector at a viscous-flow stagnation point. Phase space for the ODE divides into two attractor basins, representing particle trajectories which do or do not collide with the collector in a finite time. Written as $\ddot{x} + \dot{x} + \epsilon x^2 = 0$, we formulate the renormalisation group (RG) amplitude equations for this problem and argue that the critical trajectory which separates the attractor basins corresponds to a trivial but exact solution of these, and can therefore be extracted as a power series in $\epsilon$. We show how this can be used to find the cross-over between capture and non-capture as a function of distance from the stagnation point, for a particle released into the flow with no initial acceleration. This cross-over, previously only computable by numerical integration of the ODE, can therefore be expressed as a (numerically) convergent series with rational terms.