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

[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?

 

[Anthony]

Your statement is equivalent to

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

 

[smallphi]

Thanks!