An easy partial differential equation I have just entered the study of ODEs. However, the professor, without having talked at all about it in class, asked us to solve the following partial differential equation:
$\displaystyle r\frac{\partial v}{\partial r} = av$, where $a$ is a constant. 
Here, $v$ is a function of $r$ and $\phi$ (I transformed a harder PDE into this easier one using polar coordinates).
We are supposed to be able to use ODE methods to solve this. I cannot see how. Any help will be much appreciated!

By the method of separation of variables, let $v(r, \phi) = f(r) g(\phi)$. This implies that $\displaystyle r \frac{\partial f}{\partial r} = a f(r)$ which is a straightforward first-order linear ordinary differential equation with solution $\displaystyle f(r) = r^a + c$ and consequently $v(r, \phi) = g(\phi) (r^a + c)$
