(a) Show that the operator

$$\frac{d^{4}}{d x^{4}}+p \frac{d^{2}}{d x^{2}}+q \frac{d}{d x}+r$$

where $p(x), q(x)$ and $r(x)$ are real functions, is self-adjoint (for suitable boundary conditions which you need not state) if and only if

$$q=\frac{d p}{d x}$$

(b) Consider the eigenvalue problem

$$\frac{d^{4} y}{d x^{4}}+p \frac{d^{2} y}{d x^{2}}+\frac{d p}{d x} \frac{d y}{d x}=\lambda y$$

on the interval $[a, b]$ with boundary conditions

$$y(a)=\frac{d y}{d x}(a)=y(b)=\frac{d y}{d x}(b)=0$$

Assuming that $p(x)$ is everywhere negative, show that all eigenvalues $\lambda$ are positive.

(c) Assume now that $p \equiv 0$ and that the eigenvalue problem (*) is on the interval $[-c, c]$ with $c>0$. Show that $\lambda=1$ is an eigenvalue provided that

$$\cos c \sinh c \pm \sin c \cosh c=0$$

and show graphically that this condition has just one solution in the range $0<c<\pi$.

[You may assume that all eigenfunctions are either symmetric or antisymmetric about $x=0 .]$