A novel symbolic ordinary differential equation solver

Abstract

This paper describes a symbolic ordinary differential equation solver written as a package for the muMATH computer algebra system. The novel features of this solver are:1. Whereas previous ODE solvers generally require the equation to be quasi-linear and generally yield only implicit results, this new solver is highly integrated with an algebraic equation solver before, during and after solution. In fact, the same function named SOLVE is used for both algebraic and differential equations.2. By not insisting upon quasi-linear input, it is also practical to incorporate in the same solver methods for quasi-linear equations and a generalized version of Clairaut's equation, including singular envelope solutions. To my knowledge, previous ODE solvers have not treated even the ordinary Clairaut's equation, even using a separate solver.3. Since integration constants introduced at intermediate stages can become deeply embedded in a final explicit solution, the new solver automatically replaces subexpressions containing only an integration constant and numerical constants with a new constant, in order to greatly simplify the appearance of results, with no loss of generality.4. Whereas previous ODE solvers are generally restricted to first or second order (or to linear equations with constant coefficients) the new solver attempts to solve equations of any order, even if they are nonlinear and/or have variable coefficients.5. When the new solver is unable to obtain a closed-form solution, Picard iteration or a Taylor-series method is automatically employed so as to obtain a symbolic truncated series solution.muMATH does not yet provide the algebraic operations necessary to support the recently-discovered ODE decision procedures. Consequently the methods used to solve quasi-linear first and second order equations are essentially the same as those employed by other solvers of the pre-decision-procedure era.