2.7 Sturm–Liouville Problems 175
- Find the coefficientsbmcorresponding to
g(x)= 1 , 0 <x<a. - Using the solution of Exercise 7, write out the first few terms of the solu-
tion of Eqs. (6)–(8), whereg(x)=T,0<x<a. - Same as Exercise 7 forg(x)=x,0<x<a.
10.Verify the orthogonality integral by direct integration. It will be necessary
to use the equation that defines theλn:
κλncos(λna)+hsin(λna)= 0.
2.7 Sturm–Liouville Problems
At the end of the preceding section, we saw that ordinary Fourier series are
not quite adequate for all the problems we can solve. We can make some gen-
eralizations, however, that do cover most cases that arise from separation of
variables. In simple problems, we often find eigenvalue problems of the form
φ′′+λ^2 φ= 0 , l<x<r, (1)
α 1 φ(l)−α 2 φ′(l)= 0 , (2)
β 1 φ(r)+β 2 φ′(r)= 0. (3)It is not difficult to determine the eigenvalues of this problem and to show the
eigenfunctions orthogonal by direct calculation, but an indirect calculation is
still easier.
Suppose thatφn andφm are eigenfunctions corresponding to different
eigenvaluesλ^2 nandλ^2 m.Thatis,
φ′′n+λ^2 nφn= 0 ,φm′′+λ^2 mφm= 0 ,and both functions satisfy the boundary conditions. Let us multiply the first
differential equation byφmand the second byφn,subtractthetwo,andmove
the terms containingφnφmto the other side:
φn′′φm−φ′′mφn=(
λ^2 m−λ^2 n)
φnφm.The right-hand side is a constant (nonzero) multiple of the integrand in the
orthogonality relation
∫r
lφn(x)φm(x)dx= 0 , n=m,