Use this tag when you want to solve a linear ordinary differential equation with variable coefficients via the Frobenius method.
The Frobenius method or, Method of Frobenius, attributed to the german mathemematican Ferdinand Georg Frobenius , is a generalization of the power series method for finding an infinite series solution for a linear ordinary differential equation (ODE) with variable coefficients. In fact, when an ODE has regular singular points, power series method fails to provide enough linearly independent solutions. In such a case, Frobenius method comes to our aid.
Method:
Assume that $~x_0=0~$ is regular singular point of the differential equation $$P(x)~y''(x)+Q(x)~y'+R(x)~y=0$$
A Frobenius series (generalized Laurent series) of the form $$y=x^r~\sum_{n=0}^{\infty}a_n~x^{n}=\sum_{n=0}^{\infty}a_n~x^{n+r},\qquad a_0\neq 0 $$
can be used to solve the differential equation. The parameter$~r~$ must be chosen so that when the series is substituted into the D.E. the coefficient of the smallest power of $~x~$ is zero. This is called the indicial equation. Next, a recursive equation for the coefficients is obtained by setting the coefficient of$~x^{n+r}~$equal to zero.
Caveat: There are some instances when only one Frobenius solution can be constructed.
References:
https://en.wikipedia.org/wiki/Frobenius_method
http://mathworld.wolfram.com/FrobeniusMethod.html
http://home.iitk.ac.in/~sghorai/TEACHING/MTH203/ode14.pdf
http://www.math.mcgill.ca/gantumur/math315w14/downloads/frobenius.pdf
