Polynomial solutions for first order ODEs with constant coefficients

Abstract

Most work on finding elementary function solutions for differential equations focussed on linear equations [4, 2, 6, 1, 3]. In this paper, we try to find polynomial solutions to non-linear differential equations. Instead of finding arbitrary polynomial solutions, we will find the polynomial general solutions. For example, the general solution for (dy/dx)² - 4y = 0 is y = (x + c)², where c is an arbitrary constant. We give a necessary and sufficient condition for an ODE with constant coefficients to have polynomial general solutions. For a first order ODE of degree n and with constant coefficients, we give an algorithm of complexity O(n⁹) to decide if it has a polynomial general solution and to compute the solution if it exists.