Separation Into Differential Equations

[Chronothread]

I would use the entire template except this question is very simple and does not require all of it.

Homework Statement

How do I separate
\frac{X''(x)}{X(x)}+\frac{Y''(y)}{Y(y)}=\sigma
into ordinary differential equations when \sigma is a constant.

Thanks for your help!

 

[Pere Callahan]

You already seem to have separated the differential equation, because the x-dependence is clearly separate from the y-dependence.

On the right hand side you have two terms, one depending on x only the other depending on y only. Their sum should not depend on either x or y but should be constant.

Can you conclude HOW the x-dependent term must depend on x in order for the sum of this first term and the only(!) y-dependent second term not to depend on x?

The same for the y-dependence of the second term. Can you figure out HOW it depends on y given the fact that if you add the y-independent first term the result must be y-independent?