## Solving Linear ODEs having coefficients which are periodic or doubly-periodic functions

### George Labahn, School of Computer Science, University of Waterloo

We consider the problem of solving linear ODEs where the coefficients are periodic or doubly-periodic functions. Examples of such equations include the well-known Lame equation which comes from solving Laplace's equation in three dimensions in ellipsoidal coordinates using seperation of variables.

We give an efficient procedure for deciding if such an equation can be solved by a periodic or doubly-periodic function. In the case of second order equations we also show how find a general solution in terms of exponentials of integrals of periodic or doubly-periodic functions.