The method of Frobenius is used to develop power series solutions for differential equations. Most differential equations cannot be solved in terms of elementary functions. So often other methods such as infinite series solutions are resorted to.
The Frobenius method is very simple. Consider a differential equation(D represents d/dx):
D^2(y) + p(x)D(y) + q(x)y = 0
We assume a solution of the form
y = x^m*(a0 + a1*x +a2*x^2 + a3*x3 + ....)= sum(an*x^(m+n))
and then substitute this solution back into the original equation. To do this we note:
D(y) = sum((m+n)*an*x^(m+n-1))
When these expressions are substitued back into the differential equation it must be true that the coefficient of x^n vanishes for all n. The lowest power of x may be used to obtain an equation for m. This is called the indicial equation. Using this value of m, a relation may be found for an in terms of a(n-1), a(n-2),.... This is called the recursion relation.
Fuch's theorem gaurantees that as long as we are expanding about at worst a regular singular point, the Frobenius method will yield at least one solution. The second solution can easily be obtained once one solution is known.