Linear differential equations with rational coefficients can be found efficiently
Rational linear forms of linearizable ordinary differential equations
Symbolic Computation
Summary
The paper shows that certain complicated equations involving derivatives can always be rewritten in a simpler, linear form if their terms come from rational functions. The authors provide a way to find this simpler form without directly solving the usually hard problem of finding symmetry or transformations. Their method uses algebra and calculus tools to do this with just one parameter choice, making the process more manageable.
What this means in practice
- •For mathematical software developers: Automate transformation of high-order differential equations into linear forms with rational coefficients, enabling more robust symbolic computation tools.
- •For control systems engineers: Use the method to identify linearizable models from nonlinear system equations with rational terms, aiding system analysis and design.
A theory result. No direct application yet.
Authors
Dmitry Lyakhov
Abstract
Every scalar ordinary differential equation of order at least three with rational right-hand side that is locally linearizable by a point transformation admits a linear form with rational coefficients over the same coefficient field. We prove this by restricting the derived symmetry algebra to a coordinate line and recovering a scalar differential operator from rational symmetry-jet data. The construction uses differential elimination and linear algebra; it does not require the symmetry generators or a linearizing transformation to be solved for. A single integer parameter suffices to choose the line.