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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.

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    $\begingroup$ The characteristic equation is $r^2+\lambda=0.$ To solve it turns out to ask for the sign of the discriminant, which is $-4\lambda.$ The character of solutions (real, distinct, complex conjugate) is given by that sign. $\endgroup$ – user376343 Mar 4 at 20:35

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