= Solution
A finite point $x_0$ is an <ordinary point> when the normal-form coefficients $p,q$ are analytic at $x_0$. Then arbitrary values $y(x_0),y'(x_0)$ determine a convergent <Taylor series>, and two independent choices give two <linearly independent> local analytic solutions.
A singular point is a <regular singular point> when $(x-x_0)p(x)$ and $(x-x_0)^2q(x)$ extend analytically to $x_0$. The <Frobenius method> then provides at least one solution
$$
\boxed{y=(x-x_0)^\sigma\sum_{n=0}^{\infty}a_n(x-x_0)^n,\qquad a_0\ne0,}
$$
with a locally convergent analytic factor and an exponent fixed by the <indicial equation>. If its two <indicial roots> differ by a noninteger, there are two independent <Frobenius solutions>. For integer-separated roots a second solution may include a logarithm; a repeated root gives a <Logarithmic Frobenius solution> accompanying the single pure series. Such solutions need not be finite or ordinary <Taylor series> at the singular point.
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