Reconsider Exercise 5.3.9 with the boundary conditions Reference Exercise 5.3.9...

Reconsider Exercise 5.3.9 with the boundary conditions

Reference Exercise 5.3.9:

Consider the eigenvalue problem

(a) Show that multiplying by 1/x puts this in the Sturm–Liouville form. (This multiplicative factor is derived in Exercise 5.3.3.)

(b) Show that λ ≥ 0.

*(c) Since (5.3.10) is an equidimensional equation, determine all positive eigenvalues. Is λ = 0 an eigenvalue? Show that there is an infinite number of eigenvalues with a smallest but no largest.

(d) The eigenfunctions are orthogonal with what weight according to Sturm–Liouville theory? Verify the orthogonality using properties of integrals.

(e) Show that the nth eigenfunction has n − 1 zeros.

Reference Exercise 5.3.3:

Consider the non-Sturm–Liouville differential equation

Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm–Liouville form:

Given α(x), β(x), and γ(x), what are p(x), σ(x), and q(x)?