What Determines the Sign of Lambda in Separation of Variables?

[AStaunton]

Just quick question about sep of variables..

say have function U(x,y)=X(x)Y(y)

when do separation of variables end up with some generic case that looks like:

X''/X=Y'/Y=lamda

my question is (and I think I know now the answer but would like confirmation), is what sign should the lamda be set to to make the problem easiest to solve...

And I think that it is down to which of the two, either X or Y that are going to give us the eigenvalues...
if X gives eigen values, then set lamda to the sign that allows the X differential equation easiest to solve and vise versa if Y ODE gives eigenvalues...

I'm just looking for a rule of thumb here or any tips anyone has..Thanks

 

[fzero]

The sign of $$\lambda$$ usually depends on initial or boundary conditions. For example, we can consider solutions to

$$\frac{X''}{X} = \lambda.$$

When $$\lambda>0$$, our solutions are real exponential functions $$e^{\pm\sqrt{\lambda} x$$. It is impossible to satisfy the boundary conditions $$X(0)=X(a)=0$$ with a linear combination of these solutions.

However, if $$\lambda<0$$, we find periodic solutions $$e^{\pm i\sqrt{\lambda} x$$ (equivalent to sin and cos) for which there is a linear combination that satisfies $$X(0)=X(a)=0$$.

Therefore, you will generally need to consider the full range of parameters and decide which classes of solutions are permissible given the boundary or initial conditions.