180 Chapter 2 The Heat Equation
with boundary conditions
a. φ( 0 )=0, φ′(a)=0,
b. φ′( 0 )=0, φ(a)=0,
c. φ( 0 )=0, φ(a)+φ′(a)=0,
d. φ( 0 )−φ′( 0 )=0, φ′(a)=0,
e.φ( 0 )−φ′( 0 )=0, φ(a)+φ′(a)=0.
4.In Eqs. (1)–(3), takel=0,r=a, and show that
a. The eigenfunctions areφn(x)=α 2 λncos(λnx)+α 1 sin(λnx).
b. The eigenvalues must be solutions of the equation−tan(λa)=λ(α 1 β 2 +α 2 β 1 )
α 1 β 1 −α 2 β 2 λ^2.5.Show by applying Theorem 1 that the eigenfunctions of each of the follow-
ing problems are orthogonal, and state the orthogonality relation.
a. φ′′+λ^2 ( 1 +x)φ=0, φ( 0 )=0, φ′(a)=0;
b. (exφ′)′+λ^2 exφ=0, φ( 0 )−φ′( 0 )=0, φ(a)=0;c. φ′′+(
λ^2
x^2)
φ=0, φ( 1 )=0, φ′( 2 )=0;d. φ′′−sin(x)φ+exλ^2 φ=0, φ′( 0 )=0, φ′(a)=0.
6.Consider the problem(sφ′)′−qφ+λ^2 pφ= 0 , l<x<r,
φ(r)= 0 ,in whichs(l)=0,s(x)>0 forl<x≤r,butpandqsatisfy the conditions of
a regular Sturm–Liouville problem. Require also that bothφ(x)andφ′(x)
have finite limits asx→l+. Show that the eigenfunctions (if they exist) are
orthogonal.
7.The following problem is not a regular Sturm–Liouville problem. Why?
Solve, and show that the eigenfunctions arenotorthogonal.φ′′+λ^2 φ= 0 , 0 <x<a,
φ( 0 )= 0 ,φ′(a)−λ^2 φ(a)= 0.