Let us consider the linear homogeneous differential equation
of order
If, instead,
vanishes in a point
, this point is in general a singular point. After dividing the differential equation by
and then getting the form
some new coefficients
are continuous, and even analytic in
We introduce the so-called Frobenius method for finding solution functions in a neighbourhood of the regular singular point
, confining us to the case of a second order differential equation. When we use the quotient forms
where
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(1) |
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(2) |
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(3) |
In the Frobenius method one examines whether the equation (2) allows a series solution of the form
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(4) |
Substituting (3) and (4) to the differential equation (2) converts the left hand side to
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(5) |
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(6) |
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(7) |
The first of the equations (6) leaves
arbitrary. The next linear equations in
allow to solve successively the constants
provided that the first coefficients
,
do not vanish; this is evidently the case when the roots of the indicial equation don't differ by an integer (e.g. when the roots are complex conjugates or when
is the root having greater real part). In any case, one obtains at least for one of the roots of the indicial equation the definite values of the coefficients
in the series (4). It is not hard to show that then this series converges in a neighbourhood of the origin.
For obtaining the complete solution of the differential equation (2) it suffices to have only one solution
of the form (4), because another solution
, linearly independent on
, is gotten via mere integrations; then it is possible in the cases
that
has no expansion of the form (4).
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