Solve PDE by separation (Fourier-) method

Question

I have to solve the following pde by a separation approach: $$ x^2 u_{xx} + u_{yy} - xu_x - u = 0. $$ So I put $u(x,y) = g(x) f(y)$, substituting yields $$ x^2 g''(x) f(y) + g(x) f''(y) - x g'(x) f(y) - g(x) f(y) = 0. $$ Now I have to put this in a form were the LHS just depends on one variable, and the RHS just on the other, but I do not see a way to achieve this. So maybe there is some transformation applicable before the separation or something else. Any help?

Answer 1

You need to start factor by $f(y)$ giving the following equation: $$f(y)[x^2g''(x) - xg'(x) - g(x)] = -g(x)f''(y)$$

Now divide through by $f(y)g(x)$ and you obtain : $$\frac{x^2g''(x) - xg'(x) - g(x)}{g(x)} = \frac{f''(y)}{f(y)} = constant$$

Now you have $2$ ODEs.

Note: Separation of variables is not always possible. When I was taught this method I asked my lecturer if there was a way to know if it was possible but apparently not. The only method is try and see!