Chapter 12.23, Problem 24MP

(a) The following differential equation is often called a Sturm-Liouville equation: (is a constant parameter). This equation includes many of the differential equations of mathematical physics as special cases. Show that the following equations can be written in the Sturm-Liouville form: the Legendre equation (7.2); Bessel’s equation (19.2) for a fixed p, that is, with the parameter $\lambda$ corresponding to ${\alpha }^2;$ the simple harmonic motion equation $y^{\prime \prime }=-n^{2}y;$ the Hermite equation (22.14); the Laguerre equations (22.21) and (22.26).

(b) By following the methods of the orthogonality proofs in Sections 7 and 19, show that if $y_1$ and $y_2$ are two solutions of the Sturm-Liouville equation (corresponding to the two values ${\lambda }_1$ and ${\lambda }_2$ of the parameter $\lambda$), then $y_1$ and $y_2$ are orthogonal on $(a,b)$ with respect to the weight function $B(x)$ if .