7.22, §4 Eigenfunctions

Substitute v = T(t) X(x) into the homogeneous PDE :

Separate:

The Sturm-Liouville problem for X is now:

This is a constant coefficient ODE, with a characteristic polynomial:

The fundamentally different cases are now two real roots (discriminant positive), a double root (discriminant zero), and two complex conjugate roots (discriminant negative.) We do each in turn.

Case :

Roots k₁ and k₂ real and distinct:

X = A e^{k₁ x} + B e^{k₂ x}

Boundary conditions:

No nontrivial solutions since the roots are different.

Case :

Since k₁ = k₂ = k:

X = A e^{k x} + B x e^{k x}

Boundary conditions:

No nontrivial solutions.

Case :

For convenience, we will write the roots of the characteristic polynomial more concisely as:

where according to the solution of the quadratic

Since it can be confusing to have too many variables representing the same thing, let's agree that is our ``representative'' for b, and our ``representative'' for .In terms of these representatives, the solution is, after clean-up,

Boundary conditions:

Nontrivial solutions can only occur if

which gives us our eigenvalues, by substituting in for :

Also, choosing each B=1: