A generalized Lagrange formalism is developed for Ordinary Differential Equations (ODE) with the special function solutions¹. The formalism is based on non-standard Lagrangians, which represent a novel family of Lagrangians. It is shown that the Euler-Lagrange equation needs to be supplemented with an auxiliary condition to retrieve the original equation - this is a new phenomenon in the calculus of variations.

1. G.M. Murphy, Ordinary Differential Equations and Their Solutions, New York: Dover Publication, Inc., 2011.