Finding general solution to this pde

[climbon]

Hi, my equation is;

$$\frac{\partial}{\partial t}U(x,y,t) = 2g \left( x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x} \right) U(x,y,t)$$

I want to find the general solution to this but I don't know how to find it?

Any help would be great...thanks :D

 

[micromass]

What class is this for, a PDE class?

What have you tried already?

 

[climbon]

Yer its PDE class.

I've tried using Charactoristics, so,

$$\frac{\partial x(t)}{\partial s} = 2gy(t)$$

and

$$\frac{\partial y(t)}{\partial s} = -2gx(t)$$

With s=t. I am not sure what to do now with regards to forming a general solution, would it be something of the form,

$$U(x,y,t) = f (x_0 +2gy(t)t, y_0 -2gx(t)t, t)$$

I'm not sure how to proceed.

Thanks.

 

[fzero]

Writing the original equation in polar coordinates should be illuminating. There's a further linear change of variables that will put the equation into a form where the general solution should be obvious.