7.22, §4 Eigenfunctions

Substitute v = C(t) X(x) into the homogeneous P.D.E. :

Sturm-Liouville problem for X:

Characteristic polynomial:

Case :

Roots k₁ and k₂ real and distinct:

X = A e^{k₁ x} + B e^{k₂ x}

Boundary conditions:

No nontrivial solutions.

Case :

Since k₁ = k₂ = k:

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

Boundary conditions:

No nontrivial solutions.

Case :

Define

Boundary conditions:

Nontrivial solutions if

Choosing each B=1:

Sturm Liouville theorem: for any function F(x)

To identify the formula for the coefficients, we need to put

in standard form:

11/15/00 0:19:16 11/15/00 0:36:58 11/15/00 0:42:52 11/17/00 0:17:38 11/17/00 0:37:59