7.37, §4 Eigenfunctions

Substitute u=C(y) X(x) into the homogeneous P.D.E. u_xx + u_yy = 0:

CX'' + C'' X = 0

Sturm-Liouville problem for X:

Already solved in problem 7.19:

Characteristic polynomial:

Case :

Since

Boundary conditions:

No nontrivial solutions.

Case :

Since k₁ = k₂ = 0:

X = A + B x

Boundary conditions:

hence X=A. No undetermined constants in eigenfunctions! Get a basis by choosing any nonzero value for A. Simplest is to choose 1:

X₀(x) = 1

(Any other solution X=A can be expressed as a multiple of X₀.)

Case :

Since :

Boundary conditions:

Nontrivial solutions if

Choosing each B=1:

Total:

Sturm Liouville theorem: for any function F(x)

11/27/00 0:39:26