How can I solve this differential equation (not a obvious 2nd order ODE) I am not sure how to manipulate this 2nd order ODE to find the general solution
$$u_r u^{'}+\dfrac{u_r}{r}u=\dfrac{\nu}{r}\left(\dfrac{d}{dr}\left(r u^{'}\right) - \dfrac{u}{r}\right)$$
Where $$u_r=-\dfrac{r_0 v_w}{r}$$ where $r_0$ and $v_w$, $\nu$ are constants. I know I can rewrite the left hand side as $$\dfrac{u_r}{r}\dfrac{d}{dr}(ru)$$ but I am not sure how I can manipulate the right hand side any ideas?
Note $u(r)$ only

The usage of abbreviation $u_r$ is somewhat misleading. Instead, I would suggest to do as follows. Let us denote constant $r_0 v_w$ as $\mu$. Then your equation can be rewritten as follows:
$$
-\frac{\mu}{r^2}\left(ru'+u\right) = \frac{\nu}{r}\left(ru''+u'-\frac{u}{r}\right).
$$
After simple manipulations, we get
$$
u'' + \left(1+\frac{\mu}{\nu}\right) \frac{1}{r} u' + \left(\frac{\mu}{\nu}-1\right) \frac{1}{r^2} u = 0.
$$
This is the so-called Euler Differential Equation, cf. http://mathworld.wolfram.com/EulerDifferentialEquation.html
And general solution (which depends on $\mu$ and $\nu$) for this equation is known. 
