= Solution
For a <central potential>, <angular momentum> is conserved and the motion lies in a plane. With $L=mr^2\dot\theta$, radial <energy> conservation becomes
$$
\mathcal E=\frac12m\dot r^2+V_{\rm eff}(r),\qquad V_{\rm eff}(r)=V(r)+\frac{L^2}{2mr^2}.
$$
Allowed radii have $\mathcal E\ge V_{\rm eff}$; equality gives radial <turning points>. A well with two finite <turning points> gives regular radially bound motion. Access to arbitrarily large radii gives an unbound branch, while access to the origin can lead to collision in a singular potential. Stationary points are circular-orbit candidates; a strict local minimum gives radial <stability>. For a potential tending to zero at infinity, <energy> relative to that asymptote is useful, but a singular inner potential and centrifugal barrier must still be examined.
Integrating the angular-momentum relation with the origin of <angle> at $t=0$ gives
$$
\boxed{\theta(t)=\frac Lm\int_0^t\frac{dt'}{r(t')^2}.}
$$
For $V=-A/r+B/r^2$ at $L=L_0$, the <effective potential> is $-A/r+(L_0^2+2mB_0)/(2mr^2)$. It is exactly the Kepler <effective potential> with $L_{\rm K}=\sqrt{L_0^2+2mB_0}$, provided that expression is real. Equal radial <energy> and initial radial phase therefore give identical $r(t)$, although the angular motion differs. Writing $\psi$ for the Kepler <angle>, $\dot\theta/\dot\psi=L_0/L_{\rm K}=k$. With matched periapsis origins,
$$
\theta=k\psi,\qquad \boxed{r(\theta)=\frac{r_0}{1+e\cos(\theta/k)},\quad k=\frac{L_0}{\sqrt{L_0^2+2mB_0}}.}
$$
Here $r_0=L_{\rm K}^2/(mA)$ and the <eccentricity> is fixed by radial <energy>. For $B_0>0$, $k<1$ and successive periapses are separated by $2\pi k$, so there is retrograde apsidal motion relative to a closed Kepler ellipse. The radial equivalence is the relevant content of the <Newton's theorem of revolving orbits>.
For the pure <inverse-cube central force> set $A=0$. The free-particle reference orbit at $B=0$ is a straight line, $r=d/\cos(\psi-\psi_0)$, not a finite-$A$ conic with its old $r_0$ held fixed. To cover attractive as well as repulsive cases completely, put
$$
u=1/r,\qquad \lambda=1+\frac{2mB}{L^2},\qquad u''+\lambda u=0,\qquad \mathcal E=\frac{L^2}{2m}(u'^2+\lambda u^2),
$$
where primes denote <angle> derivatives and $L\ne0$. The <inverse-cube central orbit classification> gives the following physically positive-radius branches.
For \b[$\lambda>0$], write $u=C\cos[\sqrt\lambda(\theta-\theta_0)]$ with $C>0$, so $r=C^{-1}\sec[\sqrt\lambda(\theta-\theta_0)]$. These are scattering orbits with positive <energy> and an inner turning radius $1/C$; the angular interval between the two infinite-radius ends is $\pi/\sqrt\lambda$. Repulsion ($B>0$) gives an interval smaller than $\pi$; weak attraction ($-L^2/(2m)<B<0$) gives a larger interval and can cause winding. The $B=0$ limit is the straight line.
For \b[$\lambda=0$], $u=C+D(\theta-\theta_0)$. If $D\ne0$, reciprocal-linear spirals connect an infinite-radius end to collision at infinite accumulated <angle>. If $D=0$ and $C>0$, every radius admits a circular solution at zero <energy>: attraction exactly balances centrifugal acceleration. Each is a <marginal circular orbit under an inverse-cube force>, with no radial restoring <force>.
For \b[$\lambda<0$], put $\beta=\sqrt{-\lambda}$ and $u=C_+e^{\beta\theta}+C_-e^{-\beta\theta}$. Up to <angle> shifts and reversal, the three <energy> classes are
$$
\begin{array}{c|c|c}
\mathcal E>0&r=d\,\operatorname{csch}[\beta(\theta-\theta_0)]&\text{infinity to collision}\\
\mathcal E=0&r=d e^{-\beta(\theta-\theta_0)}&\text{logarithmic collision/escape spiral}\\
\mathcal E<0&r=d\,\operatorname{sech}[\beta(\theta-\theta_0)]&\text{outer turning radius }d
\end{array}
$$
with positive-radius domains understood. The negative-energy branch is radially bounded but reaches the singular centre on either side of its outer <turning point>; it is not a regular closed orbit. Collision requires additional physics to continue through $r=0$. Zero <angular momentum> gives radial trajectories and is excluded from the angular parametrization above.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2016/paper-1-inverse-cube-orbits.png]
{title=Original representative inverse-cube orbit branches, with force sign and critical attraction distinguished}
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