7.28, §3 Eigenfunctions

Substitute u=C(t) X(x) into the homogeneous P.D.E. u_tt = a² u_xx:

C''X = a² C X''

Sturm-Liouville problem for X:

Characteristic polynomial:

Case :

Since

Boundary conditions:

No nontrivial solutions.

Case :

Since k₁ = k₂ = 0:

X = A + B x

Boundary conditions:

No nontrivial solutions.

Case :

Since :

Boundary conditions:

Nontrivial solutions if

Choosing each B=1:

Total:

Sturm Liouville theorem:

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