Chapter 11.3, Problem 14E

Let $\lambda$ be an eigenvalue and $\varphi$ a corresponding eigenfunction for the regular Sturm-Liouville boundary value problem (1) - (2)

a. Show that $\begin{array}{l}\lambda {\displaystyle \underset{a}{\overset{b}{\int }}r\left(x\right){\varphi }^2\left(x\right)dx}\\ ={\displaystyle \underset{a}{\overset{b}{\int }}\left\{p\left(x\right){\left[{\varphi }^{\prime }\left(x\right)\right]}^2-q\left(x\right){\varphi }^2\left(x\right)\right\}dx}\\ -{\left[p\left(x\right)\varphi \left(x\right){\varphi }^{\prime }\left(x\right)\right]}_a^b\end{array}$ (38)

b. Use equation (38) to show that if $q\left(x\right)\le 0$ and $a_1=b_1=0$ in (2), then $\lambda$ is nonnegative.

c. Show that if $q\left(x\right)\le 0$ and $a_2=b_2=0$ in (2), then $\lambda$ is nonnegative.

d. Show that if $q\left(x\right)\le 0$, $\frac{a_1}{a_2}<0$ and $\frac{b_1}{b_2}>0$, then $\lambda$ is nonnegative.