When checking the eigenvalues of an ODE that you separate from a PDE like:
$\displaystyle \frac{d^2\phi}{dx^2} = -\lambda \phi$
$\phi(0)=0$
$\phi(L)=0$
Why do you separate the problem into cases depending on $\lambda$, the eigenvalue, you are trying to find?
My book does the following:
Case 1: $\lambda \gt 0$
...
Case 2: $\lambda = 0$
...
Case 3: $\lambda \lt 0$
...
Why, if you are trying to find out what $\lambda$ equals, is the procedure to set $\lambda$? It seems ciccular/counterintuitive. Like supposing what you are trying to prove?
For context, this from the chapter in a PDE book, on how to use the Method Of Separation Of Variables.