text: satisfies If we choose one of the roots to the indicial polynomial for r in , we gain a solution to the differential equation. If the difference between the roots is not an integer, we get another, linearly independent solution in the other root.
  Example
  Let us solve
  Divide throughout by z2 to give
  which has the requisite singularity at z = 0.
  Use the series solution
  Now, substituting
  From  we get a double root of 1. Using this root, we set the coefficient of  to be zero (for it to be a solution), which gives us:
  hence we have the recurrence relation:
  Given some initial conditions, we can either solve the recurrence entirely or obtain a solution in power series form.
  Since the ratio of coefficients  is a rational function, the power series can be written as a generalized hypergeometric series.
  "The exceptional cases": roots separated by an integer
  The previous example involved an indicial polynomial with a repeated root, which gives only one solution to the given differential equation.  In general, the Frobenius method gives two independent solutions provided that the indicial equation's roots are not separated by an integer (including zero).
  If the root is repeated or the roots differ by an integer, then the second solution can be found using: