I'm not sure about this, but since nobody else has replied, I'll throw in my two cents' worth.
It seems to me that this equation has two solutions. The reason for suspecting this is the naive one that you can actually derive the solution as an implicit function. In fact, . Substitute and integrate:
The initial condition tells you that the constant of integration is 0, so we get the solution as
Here's where I get a bit vague. I suspect that if you have a good enough graphing program you'll find that this curve has two branches through the origin. But I don't know of any formal way to prove that.