Transforming ODEs and PDEs with radical coefficients into rational coefficients

We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables. It also applies to systems of linear ODEs. It is based on previous work on reparametrization of radical algebraic varieties.

💡 Research Summary

This paper presents a novel algorithmic method for transforming ordinary differential equations (ODEs) and partial differential equations (PDEs) with radical (e.g., square root) function coefficients into equivalent equations with rational function coefficients. The core idea is to find a rational change of the independent variable(s)—such as x = r(z) for ODEs—that “rationalizes” the radicals appearing in the coefficients.

The theoretical foundation lies in algebraic geometry, specifically the theory of radical varieties and their reparametrization. Given a differential equation with radical coefficients, the algorithm first constructs a “radical parametrization” P = (x, a(x)), where a(x) is the tuple of all non-rational coefficients. From P, two key algebraic varieties are derived: the “radical variety” V_P and the more fundamental “tower variety” V_T. V_T essentially represents the graph of the independent variable(s) and all the radicals δ_i(x) involved.

The main theoretical result (Corollary 3.4) states that if V_T is a rational variety (i.e., can be parametrized by rational functions), then one can find a rational change of variable x = r(z) that transforms the original equation into an algebraic differential equation with rational coefficients. This rational change r(z) is obtained directly from a rational parametrization Q(z) = (r(z), δ(r(z))) of V_T. Furthermore, if Q(z) is invertible, its inverse function h allows for the reconstruction of the solution to the original equation from the solution of the transformed equation.

The paper provides a detailed algorithm (Algorithm 3.5) implementing this process for ODEs. It involves computing V_T, checking its rationality using existing algorithms, and in the affirmative case, computing the rational parametrization Q(z) and its inverse. The method is also successfully extended to systems of linear ODEs (Section 4) and conceptually to PDEs (Section 5), demonstrating its generality.

Several concrete examples are worked through, showing how equations that are intractable or yield implicit solutions in standard computer algebra systems like Maple can be transformed into rational-coefficient equations that are easily solved. The algorithm thus provides a powerful symbolic-numeric bridge, converting problems with transcendental-looking coefficients into a purely algebraic framework where a richer set of solution techniques is available.