System of differential equations from physics - Planar motion in polar coordinates.

I am trying to solve the problem of planar motion for a body in the case of a radial force, $$f(r)=-kr$$, k a constant. By considering polar coordinates $$(r,\theta)$$ for which the trajectory becomes $$(r(t),\theta(t)),t\geq0$$, the differential equations of motion are

$$m(\ddot{r}-r\dot{\theta}^2)=-kr$$ $$mr^2\dot{\theta}=L$$

(L being constant by conservation of angular momentum). Substituting the second into the first, I get $$\ddot{r}+\frac{k}{m}r=\frac{L}{r^3m^2}$$

Using a google search, I came across Bertrand's theorem (https://en.wikipedia.org/wiki/Bertrand%27s_theorem) which points to substituting $$u(\theta)=1/r(\theta)$$. Doing this rather strange substitution, I get

$$\frac{d^2u}{d\theta^2}+u=-2\frac{mk}{L^2u^3}$$

Have I done this right? If I have, then I get a nonlinear differential equation that cannot be integrated by hand... Or not?

Or is there some way to find the trajectory that I am unaware of in this case? The wiki says that for a radial force $$f=1/r$$ this would be a linear function. That I can do. This I cannot. What am I missing here?

Thanks in advance for any replies...

Continuing further $$1/r$$ has a simple harmonic motion relation with respect to $$\theta$$. This is now in the canonical Kepler/Newton form that can now include defined semi-latus rectum and eccentricity $$(p,e)$$.
$$p\cdot \frac{1}{r}= 1 - e \cos \theta$$
This $$(r-\theta)$$ integration for elliptic orbit is simpler than when persisting with $$(r-time)$$ integration in terms of elliptic integrals.