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Frobenius Solution

A Frobenius solution of a homogeneous linear second-order ordinary differential equation near a regular singular point x_0 is a solution of the form

y(x)=(x-x_0)^rsum_(n=0)^inftya_n(x-x_0)^n,
(1)

where a_0!=0 and the power series converges near x_0. The exponent r is a root of the indicial equation. The Frobenius method determines r and the coefficients, whereas the Frobenius solution is the resulting function.

The exponent need not be an integer. For example,

x^2y^('')+xy^'-1/4y=0
(2)

has the linearly independent Frobenius solutions y=x^(1/2) and y=x^(-1/2) on x>0, corresponding to the indicial equation

r^2-1/4=0.
(3)

Neither solution has a Laurent series about zero.

See also

Frobenius Method, Fuchs's Theorem, Indicial Equation, Regular Singular Point

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References

Ince, E. L. Ch. 5 in Ordinary Differential Equations. New York: Dover, 1956.National Institute of Standards and Technology. "Regular Singularities: Fuchs-Frobenius Theory." §2.7(i) in Digital Library of Mathematical Functions. https://dlmf.nist.gov/2.7#i.

Cite this as:

Weisstein, Eric W. "Frobenius Solution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FrobeniusSolution.html

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