As usual there is a snag. Most equations of interest are of a form where and/or are singular at the point (usually ). Any point where and are singular is called (surprise!) a singular point. Of most interest are a special class of singular points called regular singular points, where the differential equation can be given as
(9.10) |
with and analytic at . Let us assume that this point is . Frobenius’ method consists of the following technique: In the equation
(9.11) |
we assume a generalised series solution of the form
(9.12) |
Equating powers of we find
(9.13) |
etc. The equation for the lowest power of can be rewritten as
(9.14) |
This is called the indicial equation. It is a quadratic equation in , that usually has two (complex) roots. Let me call these , . If is not integer one can prove that the two series solutions for with these two values of are independent solutions.
Let us look at an example
(9.15) |
Here , , so is indeed a regular singular point. The indicial equation is
(9.16) |
which has roots , , which gives two independent solutions
Independent solutions:
Independent solutions are really very
similar to independent vectors: Two or more functions are
independent if none of them can be written as a combination
of the others. Thus
and
are independent, and
and
are dependent.