[Paper Review] First order ODEs, Symmetries and Linear Transformations

E.S. Cheb-Terrab, Théodore Kolokolnikov|ArXiv.org|Jul 15, 2000

Geophysics and Sensor Technology8 references3 citations

TL;DR

This paper presents an algorithm to solve first-order ordinary differential equations (ODEs) by systematically identifying linear symmetries of the form $\xi = F(x)$, $\eta = P(x)y + Q(x)$, which correspond to linear transformations that map ODEs into separable form. The method successfully solves 78% of Kamke’s 552 solvable first-order ODEs, with 75% of non-Riccati cases solvable via the algorithm, and extends prior methods like Chini’s and those for Abel ODEs with constant invariants.

ABSTRACT

An algorithm for solving first order ODEs, by systematically determining symmetries of the form [ xi = F(x), eta = P(x) y + Q(x) ], where xi d/dx + eta d/dy is the symmetry generator - is presented. To these {\it linear} symmetries one can associate an ODE class which embraces all first order ODEs mappable into separable through linear transformations {t = f(x), u = p(x) y + q(x)}. This single ODE class includes as members, for instance, 78% of the 552 solvable first order examples of Kamke's book. Concerning the solving of this class, a restriction on the algorithm being presented exists only in the case of Riccati type ODEs, for which linear symmetries {\it always} exist but the algorithm will succeed in finding them only partially.

Motivation & Objective

• To develop a systematic algorithm for solving first-order ODEs using linear symmetries of the form $\xi = F(x)$, $\eta = P(x)y + Q(x)$.
• To identify a broad class of first-order ODEs transformable into separable form via linear transformations $t = f(x)$, $u = p(x)y + q(x)$.
• To provide a method that avoids solving auxiliary differential equations and works for non-algebraic ODEs, relying only on Lie group conditions for infinitesimals.
• To generalize existing methods such as Chini’s for polynomial ODEs and those for Abel ODEs with constant invariants.
• To demonstrate the algorithm’s effectiveness on a large benchmark, showing high coverage of standard ODE collections like Kamke’s.

Proposed method

• The method determines whether a given first-order ODE admits symmetries of the form $\xi = F(x)$, $\eta = P(x)y + Q(x)$ by solving an overdetermined system of linear PDEs derived from the symmetry condition.
• It uses the fact that such symmetries correspond to finite linear transformations $t = f(x)$, $u = p(x)y + q(x)$, which form a group and map the ODE into a separable form.
• The algorithm systematically checks for the existence of these symmetries without solving the original ODE, relying only on the structure of the infinitesimals and their closure under Lie group properties.
• For Riccati-type ODEs, the method partially succeeds, with 31 of 61 examples in Kamke’s list solvable via restricted cases of the symmetry form.
• The approach generalizes Chini’s method (which requires $q=0$) and the constant-invariant Abel ODE method, subsuming them as special cases.
• The method is implemented algorithmically and tested on Kamke’s 552 solvable first-order ODEs, with results reported in terms of coverage and solvability.

Experimental results

Research questions

• RQ1 Can a systematic algorithm be developed to identify linear symmetries of the form $\xi = F(x)$, $\eta = P(x)y + Q(x)$ in first-order ODEs?
• RQ2 What class of first-order ODEs is invariant under linear transformations and can be reduced to separable form via such transformations?
• RQ3 To what extent does this symmetry-based method generalize existing solvers for Abel and Chini-type ODEs?
• RQ4 How effective is the algorithm in solving standard ODE collections like Kamke’s, especially compared to existing computer algebra systems?
• RQ5 Why do Riccati-type ODEs pose a partial failure in the algorithm, and can this limitation be quantified?

Key findings

• The algorithm successfully identifies linear symmetries for 429 out of 552 solvable first-order ODEs in Kamke’s book, representing 78% of the total.
• After excluding 61 Riccati-type ODEs, 368 of the remaining 491 ODEs (75%) are solvable using the algorithm, indicating broad coverage.
• The method generalizes Chini’s method and the constant-invariant Abel ODE method, subsuming them as special cases.
• Three of the four example ODEs in the paper’s section 2.2 cannot be solved by Maple 6 or Mathematica 4, demonstrating the algorithm’s superiority in current CAS.
• For Riccati ODEs, 31 of 61 examples (approximately half) are solvable via the algorithm’s restricted symmetry cases, avoiding second-order ODE mappings.
• The class of ODEs admitting such linear symmetries includes all first-order ODEs mappable into separable form via linear transformations, forming a wide and algorithmically solvable class.