Fuchs' theorem explained

In mathematics, Fuchs' theorem, named after Lazarus Fuchs, states that a second-order differential equation of the form$$y$$ + p(x)y' + q(x)y = g(x)has a solution expressible by a generalised Frobenius series when

p(x) , q(x) and g(x) are analytic at x = a or a is a regular singular point. That is, any solution to this second-order differential equation can be written as$$y\; =\; \backslash sum\_^\backslash infty\; a\_n\; (x\; -\; a)^,\; \backslash quad\; a\_0\; \backslash neq\; 0$$for some positive real s, or$$y\; =\; y\_0\; \backslash ln(x\; -\; a)\; +\; \backslash sum\_^\backslash infty\; b\_n(x\; -\; a)^,\; \backslash quad\; b\_0\; \backslash neq\; 0$$for some positive real r, where y₀ is a solution of the first kind.

Its radius of convergence is at least as large as the minimum of the radii of convergence of

p(x) , q(x) and g(x) .

See also

• Frobenius method

References

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