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I have the following differential equation resulting from a simplification of Navier-Stokes in cylindrical coordinates. $$ \frac{d^2 u_{\theta}}{d r^2}+\frac{d}{dr} \bigg(\frac{u_{\theta}}{r}\bigg)=0 $$ where $u_{\theta}$ is only a function of $r$. This is actually a simplified version using a reverse chain rule for the second term which I thought could help. Does this have a solution and how would you go about solving it?

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$$ \frac{d^2 u}{d r^2}+\frac{d}{dr} \bigg(\frac{u}{r}\bigg)=\frac{d^2 u}{d r^2}+\frac{1}{r}\frac{du}{dr}-\frac{1}{r^2}u=0 $$ Particular solutions on the form $y=r^k$ $$k(k-1)r^{k-2}+\frac{1}{r}kr^{k-1}-\frac{1}{r^2}r^k \quad\to\quad k(k-1)+k-1=0 \quad\to\quad k=\pm 1$$ The two independent solutions $r^1$ and $r^{-1}$ lead to the general solution : $$u=c_1r+c_2\frac{1}{r}$$

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    That works! Thanks a lot. So how did you know to assume $y=r^k$ for the P.I. ? Would you recommend any kind of handbook for this kind of stuff? We get a lot of analytical simplifications of this sort and although we did common PDEs using sep of variables, this kind of 'simple' ODEs are always the challenging ones. – 2017-02-06
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    They are some well known kind of differential equations which have to be studied at early stage of learning. This one is the Cauchy–Euler equation. https://en.wikipedia.org/wiki/Cauchy%E2%80%93Euler_equation – 2017-02-06