Frobenius method

The Frobenius method, after Ferdinand Georg Frobenius, is a method to solutions of the ordinary differential equation

To find, be provided where and as analytic in a neighborhood of. The idea is solutions in the form of a generalized power series

To be set and to determine the unknown coefficients by comparing coefficients. The central theorem was first proved by Lazarus Immanuel Fuchs based on the work of Karl Weierstrass and then generalized by Frobenius.

Set of Fuchs

Without loss of generality we can set. Consider the differential equation

With a maximum at 0 first-order pole at 0 and a pole at most second order. So you can in the form

Be written, where the series converge in a neighborhood of 0.

The characteristic exponents

The solutions of the characteristic equation

And we can arrange it according.

Then the following case distinction applies:

• Is not an integer, then there exist two solutions of the form

• Is an integer, then there exist two solutions of the form

The radius of convergence corresponds to the minimum of the radius of convergence of the series for and.

The converse is also true: If there are two solutions of the above form, at 0 has a pole of first order maximum at 0 and a pole at most second order.

A differential equation with meromorphic coefficients are for all singularities ( inclusive) of the above type is called a Fuchsian differential equation.

Generalizations

The set of Fuchs can be generalized to higher-order differential equations and systems of differential equations of first order.

Examples

Some examples that can be solved by the method of Frobenius:

• Bessel's differential equation
• Legendre's differential equation
• Laguerre'sche differential equation
• Hypergeometric differential equation