Frobenius Method

If $x_0$ is an ordinary point of the Ordinary Differential Equation, expand $y$ in a Taylor Series about $x_0$, letting

$$y = \sum_{n=0}^\infty a_nx^n.$$ (1)

Plug $y$ back into the ODE and group the Coefficients by Power. Now, obtain a Recurrence Relation for the $n$th term, and write the Taylor Series in terms of the $a_n$s. Expansions for the first few derivatives are

$$y = \sum_{n=0}^\infty a_nx^n$$ (2)

$$y' = \sum_{n=1}^\infty na_nx^{n-1} = \sum_{n=0}^\infty (n+1)a_{n+1}x^n$$ (3)

$$y'' = \sum_{n=2}^\infty n(n-1)a_nx^{n-2} = \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^n.$$ (4)

If $x_0$ is a regular singular point of the Ordinary Differential Equation,

$$P(x)y''+Q(x)y'+R(x)y = 0,$$ (5)

solutions may be found by the Frobenius method or by expansion in a Laurent Series. In the Frobenius method, assume a solution of the form

$$y = x^k\sum_{n=0}^\infty a_nx^n,$$ (6)

so that

$$y = x^k \sum_{n=0}^\infty a_nx^n = \sum_{n=0}^\infty a_nx^{n+k}$$ (7)

$$y' = \sum_{n=0}^\infty a_n(n+k)x^{k+n-1}$$ (8)

$$y'' = \sum_{n=0}^\infty a_n(n+k)(n+k-1)x^{k+n-2}.$$ (9)

Now, plug $y$ back into the ODE and group the Coefficients by Power to obtain a recursion Formula for the $a_n$th term, and then write the Taylor Series in terms of the $a_n$s. Equating the $a_0$ term to 0 will produce the so-called Indicial Equation, which will give the allowed values of $k$ in the Taylor Series.

Fuchs's Theorem guarantees that at least one Power series solution will be obtained when applying the Frobenius method if the expansion point is an ordinary, or regular, Singular Point. For a regular Singular Point, a Laurent Series expansion can also be used. Expand $y$ in a Laurent Series, letting

$$y = c_{-n}x^{-n}+\ldots + c_{-1}x^{-1}+c_0+c_1x+\ldots + c_nx^n+\ldots.$$ (10)

Plug $y$ back into the ODE and group the Coefficients by Power. Now, obtain a recurrence Formula for the $c_n$th term, and write the Taylor Expansion in terms of the $c_n$s.

See also Fuchs's Theorem, Ordinary Differential Equation

References

Arfken, G. ``Series Solutions--Frobenius' Method.'' §8.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 454-467, 1985.