I encountered the following $2^{\text{nd}}$-order, nonlinear ODE while working on a classical mechanics problem:
$$ \frac{d^2r}{dt^2}-\frac{\alpha^2}{r^3}+\beta=0 $$
where $\alpha, \beta > 0$ are given.
I'm really struggling to find an analytic solution. I tried taking the inverse Laplace Transform, but when I transformed back, I got nonsense. Any advice on a good strategy for solving this problem?
Motivation
Suppose you have a marble constrained to move along the inside surface of a cylindrical, frictionless, concave cone. Its Lagrangian is given by
$$L=\frac{m}{2}\left(\dot r^2+(r\dot\phi)^2+\dot z^2\right) -mgz+\lambda(z-r\tan(\theta))$$
where $(r,\phi,z)$ are the normal cylindrical coordinates, $\theta$ is the inclination of the cone above horizontal, and $\lambda$ is the Lagrange multiplier. Solving Lagrange's equations to eliminate $\lambda$ and rewrite $z$ in terms of $r$ gives the above equation.
Removing the $z$-coordinate with the constraint leads to the reduced Lagrangian
$$\tag{1} L~=~ \frac{\mu}{2}\dot{r}^2 +\frac{m}{2}r^2\dot{\phi}^2 -V(r),$$
where $\mu:=\frac{m}{\cos^2\theta}$ and $V(r):=mgr\tan\theta $. The canonical momenta are
$$ \tag{2} p_r~=~ \mu\dot{r}, \qquad p_{\phi}~=~mr^2\dot{\phi}. $$
Performing the Legendre transformation leads to Hamiltonian
$$ \tag{3} H~=~ \frac{p_r^2}{2\mu}+\frac{p_{\phi}^2}{2mr^2} +V(r). $$
The angular momentum $p_{\phi}$ is a constant of motion. A first integral is given by energy conservation
$$ \tag{4} E~=~ \frac{\mu}{2}\dot{r}^2 + \frac{p_{\phi}^2}{2mr^2}+V(r), $$
which leads to
$$ \tag{5} \Delta t ~=~\pm\int \frac{dr}{ \sqrt{\frac{2}{\mu}(E-\frac{p_{\phi}^2}{2mr^2}-V(r))}} .$$
Eq. (5) agrees up to signs with the integral that OP writes in a comment. We note that all orbits are radially bound. We know from Bertrand's theorem that not all radially bound orbits are closed.