Worked example - series solution of y'' - 2x y' - 4y = 0 about x = 0 (ordinary point; this is a Hermite-type equation). Try y = sum(n>=0) a_n x^n. Then y' = sum n a_n x^{n-1}, y'' = sum n(n-1) a_n x^{n-2}. Substitute: sum n(n-1) a_n x^{n-2} - 2x sum n a_n x^{n-1} - 4 sum a_n x^n = 0. Shift index on the first sum (m = n-2): sum(m>=0) (m+2)(m+1) a_{m+2} x^m - sum 2n a_n x^n - sum 4 a_n x^n = 0. Recurrence: a_{n+2} = 2(n+2) a_n/((n+2)(n+1)) = 2 a_n/(n+1). With a_0 = 1, a_1 = 0: a_2 = 2, a_4 = 2x2/3 = 4/3, ... even series; with a_0 = 0, a_1 = 1: a_3 = 2/2 = 1, a_5 = 2/4 = 1/2, ... odd series. Two independent solutions built term by term - the Hermite functions in embryo.