# The ONLY solution of PDE: f_t g_r = f_r g_t ?

1. Jul 5, 2008

### smallphi

The ONLY solution of PDE: f_t g_r = f_r g_t ???

I have the following PDE:

$$\frac{\partial f(r,t)}{\partial t} \, \, \frac{\partial g(r,t)}{\partial r} = \frac{\partial f(r,t)}{\partial r} \, \, \frac{\partial g(r,t)}{\partial t}$$

By a simple check, I know a solution is f = h(g), where h() is arbitrary function. The Maple PDE solver returns exactly that.

How can I prove, f=h(g) is the ONLY type of solution of that PDE?

Last edited: Jul 5, 2008
2. Jul 8, 2008

### Anthony

Re: The ONLY solution of PDE: f_t g_r = f_r g_t ???

$$\frac{\partial (f,g)}{\partial (t,\tau)} = 0$$