Let us consider the linear homogeneous differential equation
of order (http://planetmath.org/DifferentialEquation) . If the coefficient functions are continuous and the coefficient of the highest order derivative (http://planetmath.org/HigherOrderDerivatives) does not vanish on a certain interval (resp. a domain (http://planetmath.org/Domain2) in ), then all solutions are continuous on this interval (resp. ). If all coefficients have the continuous derivatives up to a certain , the same concerns the solutions.
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 (http://planetmath.org/DifferentialEquation) differential equation. When we use the quotient (http://planetmath.org/Division) forms
where , and are analytic in a neighbourhood of and , our differential equation reads
Since a change of variable brings to the case that the singular point is the origin, we may suppose such a starting situation. Thus we can study the equation
where the coefficients have the converging power series expansions
In the Frobenius method one examines whether the equation (2) allows a series solution of the form
where is a constant and .
Substituting (3) and (4) to the differential equation (2) converts the left hand to
Our equation seems clearer when using the notations :
Thus the condition of satisfying the differential equation by (4) is the infinite system of equations
In the first , since , the indicial equation
must be satisfied. Because , this quadratic equation determines for two values, which in special case may coincide.
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 (http://planetmath.org/Equation) of the indicial equation don’t differ by an integer (e.g. when the are complex conjugates or when is the having greater real part). In any case, one obtains at least for one of the 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 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).
- 1 Pentti Laasonen: Matemaattisia erikoisfunktioita. Handout No. 261. Teknillisen Korkeakoulun Ylioppilaskunta; Otaniemi, Finland (1969).
|Date of creation||2013-03-22 17:43:49|
|Last modified on||2013-03-22 17:43:49|
|Last modified by||pahio (2872)|
|Synonym||method of Frobenius|