Solving a Non-linear Differential Equation: Help Needed

[Benny]

Hi, can someone please help me with the following differential equation? I need to find the general solution.

$$x\frac{{dy}}{{dx}} = x^2 - y^2$$

It's non-linear so I didn't bother with rearranging the equation. It doesn't look seperable either so that doesn't really leave me with much to go on with the knowledge that I have. Since the basic techniques were not applicable I tried differentiating both sides wrtx and other things like that to see if I could get the equation into a form which is easier to work with. That didn't get me anywhere so could someone help me out?

There might be something simple that I'm missing, after all it took me a few days to remember that y' = (y)^2 is solvable by separation of variables () so even a small suggestion would be helpful.

 

[Data]

This does not have a nice solution. What do you need this for? It's an inexact equation with no single-variable integrating factor. The solutions Maple finds are in terms of several Bessel functions.

 

[Benny]

I thought that there would be a fairly neat solution. The question asks for the exact general solution and then asks for a phase diagram of some system so I expected a 'nice' answer. Thanks anyway.