Discussion
For the differential equation
\[ \begin{aligned} &x^{2}\frac{d^{2}y}{dx^{2}} + x\frac{dy}{dx} \\ &+ (x^{2} - n^{2})y = 0 \end{aligned} \]
Which of the following statements is true ?
Answer: C
\(x = 0\) is a regular singular point, while the point at infinity is an irregular singular point.
This is Bessel's equation. Dividing by \(x^{2}\) gives \(y'' + P(x)y' + Q(x)y = 0\) with \(P = \frac{1}{x}\) and \(Q = 1 - \frac{n^{2}}{x^{2}}\). Both are singular at 0, but \(xP = 1\) and \(x^{2}Q = x^{2} - n^{2}\) are analytic there, so \(x = 0\) is a regular singular point (Frobenius series work).
At infinity, substitute \(x = 1/t\): the transformed equation has a coefficient behaving like \(1/t^{4}\) at \(t = 0\), too strong a singularity to be regular, so infinity is an irregular singular point. That is why Bessel functions decay like \(\frac{\cos x}{\sqrt{x}}\) with an essential oscillation rather than a power law. The options calling \(x = 0\) ordinary are wrong because \(P\) blows up there.
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