91Ó°ÊÓ

Find all values of \(\alpha\) for which all solutions of \(x^{2} y^{\prime \prime}+\alpha x y^{\prime}+(5 / 2) y=0\) approach zero as \(x \rightarrow \infty\).

The definitions of an ordinary point and a regular singular point given in the preceding sections apply only if the point \(x_{0}\) is finite. In more advanced work in differential equations it is often necessary to discuss the point at infinity. This is done by making the change of variable \(\xi=1 / x\) and studying the resulting equation at \(\xi=0 .\) Show that for the differential equation \(P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0\) the point at infinity is an ordinary point if $$ \frac{1}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi}-\frac{Q(1 / \xi)}{\xi^{2}}\right] \quad \text { and } \quad \frac{R(1 / \xi)}{\xi^{4} P(1 / \xi)} $$ have Taylor series expansions about \(\xi=0 .\) Show also that the point at infinity is a regular singular point if at least one of the above functions does not have a Taylor series expansion, but both \(\frac{\xi}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi}-\frac{Q(1 / \xi)}{\xi^{2}}\right] \quad\) and \(\quad \frac{R(1 / \xi)}{\xi^{2} P(1 / \xi)}\) do have such expansions.

The equation $$y^{\prime \prime}-2 x y^{\prime}+\lambda y=0, \quad-\infty

Use the results of Problem 21 to determine whether the point at infinity is an ordinary point, a regular singular point, or an irregular singular point of the given differential equation. \(y^{\prime \prime}+y=0\)