the method of Frobenius
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The plain power-series method works beautifully at an ordinary point, but it falls apart at a singular point — and singular points are often exactly where the physics happens. The method of Frobenius is the rescue: a clever extension that finds series solutions at a regular singular point by allowing the series to start with a non-integer power out front. It is the tool that gives us Bessel functions, and it earns its keep precisely where the simple method gives up.
Here is the trick in plain steps. Instead of y = sum a_n x^n, you try a 'Frobenius series' y = x^r times (a_0 + a_1 x + a_2 x^2 + ...), where r is an unknown exponent (possibly a fraction, possibly negative) that you must determine, and a_0 is taken nonzero. You substitute this into the equation. The very lowest power of x produces an equation for r alone — the indicial equation — whose roots tell you the allowed exponents. The remaining powers give a recurrence for the a_n just as before. The leading factor x^r is exactly what absorbs the mild singularity that a plain Taylor series could not handle.
The full method has cases depending on how the two roots of the indicial equation relate, and sometimes a logarithm has to be added to capture the second solution — but the core idea is simply 'Taylor series with a flexible power in front.' Its great limitation is honest: it is guaranteed only at regular singular points. At an irregular singular point the leading-power dodge is not enough, and you must look elsewhere.
For 2x y'' + y' + y = 0 at the regular singular point x = 0, try y = x^r (a_0 + a_1 x + ...). The lowest power gives the indicial equation 2 r(r-1) + r = 0, i.e. r(2r - 1) = 0, so r = 0 or r = 1/2. The half-integer exponent r = 1/2 is exactly what a plain Taylor series could never produce.
The exponent r is fractional here (1/2) — the leading x^(1/2) is the heart of why Frobenius reaches where ordinary series cannot.
Always require a_0 nonzero; that is what fixes r as the true leading exponent. If you accidentally let a_0 = 0 you have merely shifted into a higher power and learned nothing about the indicial roots.