Chapter 8

Show that the Euler-Lagrange equation derived from any integrand of the form. $$ f=q(x) \phi^{\prime 2}+2 r(x) \phi \phi^{\prime}+p(x) \phi^{2}+\frac{d}{d x} g(x, \phi) $$ is self-adjoint.

Prove the nondegeneracy of the Sturm-Liouville eigenvalues-that there exists only one linearly independent eigenfunction to each eigenvalue, that is. HINT: Compare exercise 3, Chap. 7 , but show that \(\tau w=\) constant \(=\) zero.