Solve by Separation of Variables:

d^2/dx^2 + 1/x(du/dx) + 1/x^2(d^2u/dy^2)=0

Question

Solve by Separation of Variables: 

d^2/dx^2 + 1/x(du/dx) + 1/x^2(d^2u/dy^2)=0

fwizbang · Answer

Assume u(x,y)= f(x)g(y), and plug this back into the DEQ

anonymous · Answer

Is this supposed to be$$u_{xx}+\frac{1}{x}u_x+\frac{1}{x^2}u_{yy}=0?$$

anonymous · Answer

abstracted: yes, I am just lazy and wrote it that way :/

anonymous · Answer

Start with what fwiz said, i'll give you the first couple steps

You're assuming that the solution is of the form

$$u(x,y)=f(x)g(y)$$

So you will plug this into the orginal equation and get:
$$f''(x)g(y)+\frac{1}{x}f'(x)g(y)+\frac{1}{x^2}f(x)g''(y)=0$$

The point of separation of variables is to get all the functions of each variable on their own side, do you know what to do next?

I just started this class too, it can be quite a handful :/

anonymous · Answer

To be honest, I'm not exactly sure what to do next. And yes it can haha....

anonymous · Answer

You're gonna want to divide both sides by something so that each term only has f(x) functions or g(x) functions. You want to do this so you can group all the f(x) on one side and the g(y) on the other.

anonymous · Answer

Ohhhhh gotcha, this is an old problem and I completely forgot how to do it... but that refreshed my memory. Here I come final! Lol, but thanks abstracted!

anonymous · Answer

Your welcome!