# Checking Eigenvalues Of an ODE

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.

• 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. – user376343 Mar 4 at 20:35