method of Frobenius
/ Frobenius: fro-BAY-nee-us /
At an ordinary point a plain power series solves the equation; at a regular singular point that plain series usually fails, because the solution may blow up or carry a fractional power. The method of Frobenius is the clever upgrade: assume a power series multiplied by an unknown power (x - x0)^r, and let the equation tell you what r must be. It rescues series solutions exactly at the points where the equations of physics are most interesting.
Around a regular singular point (take x0 = 0) you posit y = x^r times sum over k>=0 of a_k x^k, with a_0 not zero and the exponent r unknown. Substitute into the equation and look at the very lowest power of x: forcing its coefficient to vanish gives the indicial equation, a quadratic in r whose two roots r1 and r2 are the allowed leading exponents. For each root you then get a recurrence relation determining the a_k. The two roots usually give two independent solutions; but when r1 and r2 are equal, or differ by an integer, the second solution can require an extra logarithm term — Frobenius supplies the recipe for that case too.
This single method generates the named functions of mathematical physics. Bessel's equation has a regular singular point at the origin; Frobenius there yields the Bessel functions, including the J of fractional order. The hypergeometric and confluent hypergeometric equations are solved the same way. Whenever a problem in a cylinder, a sphere, or a quantum well produces a singular point at the axis or origin, Frobenius is the tool that builds the physically meaningful solution that stays finite there.
For 2x y'' + y' + y = 0 (regular singular point at 0), try y = x^r sum a_k x^k. The lowest power gives the indicial equation 2r(r-1) + r = 0, i.e. r(2r - 1) = 0, so r = 0 or r = 1/2. The two roots, differing by a non-integer 1/2, give two independent Frobenius solutions, one starting like x^{1/2} — a fractional power no ordinary Taylor series could produce.
The exponent r, not just the coefficients, is determined by the equation.
Frobenius applies only at a regular singular point. At an irregular one it fails (the indicial machinery breaks down), and you need asymptotic methods such as WKB instead. Also: when the two roots are equal or differ by an integer, do not assume two clean series solutions — check whether a logarithm is forced.