Closed form solutions of linear odes having elliptic function coefficients

Authors:
Reinhold Burger;George Labahn;Mark van Hoeij
Affiliations:
University of Waterloo, Waterloo, Ontario, Canada;University of Waterloo, Waterloo, Ontario, Canada;Florida State University, Tallahassee, Florida
Venue:
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Year:
2004

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Abstract

We consider the problem of finding closed form solutions of linear differential equations having coefficients which are elliptic functions. For second order equations we show how to solve such an ode in terms of doubly periodic functions of the second kind. The method depends on two procedures, the first using a second symmetric power of an ode along with a decision procedure for determining when such equations have elliptic function solutions while the second involves the computation of exponential solutions.