indicial equation
When you attempt a Frobenius solution y = x^r times a power series at a regular singular point, the first thing you must learn is the exponent r — the leading power with which the solution emerges from the singularity. The indicial equation is the small algebraic equation that pins down the allowed values of r. It is the gatekeeper of the Frobenius method: solve it first, and everything else follows.
You get it by substituting the Frobenius ansatz into the differential equation and examining the very lowest power of x. Because a_0 is non-zero by assumption, the coefficient of that lowest power must vanish on its own, and that condition is a quadratic in r: typically r(r - 1) + p0 r + q0 = 0, where p0 and q0 are the leading values of x P(x) and x^2 Q(x) at the singular point. Its two roots r1 and r2 (taken with r1 >= r2) are called the indicial roots or exponents. They tell you the leading behaviour x^{r1} and x^{r2} of the two solutions before you compute a single series coefficient.
The relationship between the two roots decides the whole character of the solution set. If r1 - r2 is not an integer, you get two independent honest Frobenius series. If r1 = r2, the two roots coincide and the second solution must contain a logarithm. If r1 - r2 is a positive integer, the second solution may or may not need a logarithm — you have to check. This trichotomy is exactly why Bessel functions of integer order come paired with a second solution Y that carries a logarithmic singularity, the mathematical fingerprint of equal-or-integer-separated indicial roots.
Bessel's equation of order n has, at the origin, the indicial equation r^2 - n^2 = 0, so r = +n and r = -n. The two roots differ by 2n. When n is not a half of an odd-or-even split they give independent solutions J of +n and -n order; when n is an integer the roots differ by an integer and the second solution must be the logarithm-bearing Y.
The indicial roots +n, -n explain why integer-order Bessel functions need a logarithmic partner.
The indicial equation depends only on the leading (lowest-order) behaviour of the coefficients at the singular point, not on the whole equation. So you can read the exponents off p0 = lim x P(x) and q0 = lim x^2 Q(x) immediately, without doing the full series.