Problem 4.7.1. Reduce each of the following to Sturm–Liouville form, identifying \(p\), \(q\), \(w\) and the eigenvalue in each case:

(a)

Chebyshev’s equation of the first kind, \(\left (1-x^{2}\right )y''-x\,y'+n^{2}y=0\);

(b)

Laguerre’s equation, \(x\,y''+(\alpha +1-x)\,y'+n\,y=0\);

(c)

Chebyshev’s equation of the second kind, \(\left (1-x^{2}\right )y''-3x\,y'+n(n+2)y=0\).

Where to start: Theorem 4.2.1; compute \(\int a_1/a_2\) in each case and read off \(p\) and \(\mu \).

Problem 4.7.2. Verify that the boundary term (4.2) vanishes for Legendre’s equation on \([-1,1]\), and explain why no boundary condition beyond boundedness needs to be imposed. Where to start: \(p(x)=1-x^{2}\) vanishes at both endpoints.

Problem 4.7.3. Show that the trigonometric system \(\left \{\sin \left (n\pi x/L\right )\right \}^{\infty }_{n=1}\) consists of eigenfunctions of the Sturm–Liouville problem \(y''+\lambda y = 0\) on \([0,L]\) with \(y(0)=y(L)=0\), identify the eigenvalues, and deduce the orthogonality relation from Theorem 4.3.2 rather than by direct integration.

Problem 4.7.4. Prove that eigenfunctions of a regular Sturm–Liouville problem corresponding to the same eigenvalue are linearly dependent; that is, the eigenvalues are simple. Where to start: two solutions with the same \(\lambda \) both satisfy the boundary condition at \(a\), so their Wronskian vanishes there; Abel’s identity gives \(p\,W = \text {constant}\).

Problem 4.7.5. The Rayleigh quotient for a Sturm–Liouville problem is \[R[y] = \frac {\displaystyle -\Big [p\,y\,y'\Big ]^{b}_{a} + \int ^{b}_{a}\left (p\,y'^{2}-q\,y^{2}\right )dx} {\displaystyle \int ^{b}_{a}y^{2}\,w\,dx}.\] Show that if \(y\) is an eigenfunction with eigenvalue \(\lambda \) then \(R[y]=\lambda \), and deduce that if \(q\leq 0\) and the boundary term vanishes then every eigenvalue is non-negative. Where to start: multiply (4.1) by \(y\), integrate over \([a,b]\), and integrate the first term by parts.

Problem 4.7.6. Explain why the orthogonality proofs of Chapter 3, each carried out separately by repeated integration by parts of a Rodrigues formula, are all instances of Theorem 4.3.2. What does each of those proofs gain, if anything, over the general argument?