Exercise 102

Put the following problems into the standard form for Sturm--Liouville problems, that is, find $p(x)$, $q(x)$, $r(x)$, $\alpha_1$, $\alpha_2$, $\beta_1$, and $\beta_2$, and decide if the problems are regular or not.

1. $x y'' + \lambda y = 0$ for $0 < x < 1$, $y(0) = 0$, $y(1) = 0$.
2. $(1+x^2) y'' + 2xy' + (\lambda-x^2) y = 0$ for $-1 < x < 1$, $y(-1) = 0$, $y(1)+y'(1) = 0$.