A systematic method of finding linearizing transformations for nonlinear ordinary differential equations: I. Scalar case

In this set of papers we formulate a stand alone method to derive maximal number of linearizing transformations for nonlinear ordinary differential equations (ODEs) of any order including coupled ones from a knowledge of fewer number of integrals of motion. The proposed algorithm is simple, straightforward and efficient and helps to unearth several new types of linearizing transformations besides the known ones in the literature. To make our studies systematic we divide our analysis into two parts. In the first part we confine our investigations to the scalar ODEs and in the second part we focuss our attention on a system of two coupled second order ODEs. In the case of scalar ODEs, we consider second and third order nonlinear ODEs in detail and discuss the method of deriving maximal number of linearizing transformations irrespective of whether it is local or nonlocal type and illustrate the underlying theory with suitable examples. As a by-product of this investigation we unearth a new type of linearizing transformation in third order nonlinear ODEs. Finally the study is extended to the case of general scalar ODEs. We then move on to the study of two coupled second order nonlinear ODEs in the next part and show that the algorithm brings out a wide variety of linearization transformations. The extraction of maximal number of linearizing transformations in every case is illustrated with suitable examples.

💡 Research Summary

The paper presents a unified, algorithmic framework for constructing the maximal set of linearizing transformations for nonlinear ordinary differential equations (ODEs) of arbitrary order, including coupled systems, using only a limited number of known integrals of motion. The authors divide the study into two main parts.

In the first part they focus on scalar ODEs. Starting with a second‑order nonlinear equation, they show how a single first integral (I_1(t,x,\dot{x})) can be employed to generate both local (point or contact) and non‑local transformations. By redefining the independent variable as a function of (I_1) and the dependent variable as either (I_1) itself or another function of it, the original nonlinear equation is reduced to a linear second‑order equation or even to the trivial equation (u’’=0). The authors introduce a “direct integral transformation,” a new type that substitutes the integral directly for the new independent variable, extending the range of admissible transformations beyond the classical ones.

The analysis is then extended to third‑order scalar ODEs. Two independent first integrals (I_1) and (I_2) are obtained, and the authors construct three families of transformations: (i) a simple two‑integral mapping ((\tau=I_1,,u=I_2)) that yields a linear second‑order equation; (ii) a “multi‑integral chain transformation,” where successive integrals are integrated and substituted to produce a non‑local change of variables that can collapse a third‑order nonlinear equation directly to a first‑order linear equation; and (iii) mixed transformations that combine one integral as the new independent variable and the other as the new dependent variable. The multi‑integral chain transformation is highlighted as a novel contribution not previously reported in the literature.

For a general (n)‑th order scalar ODE the method scales naturally: with (n-1) independent first integrals one can construct up to (n) distinct linearizing maps. The authors provide a Jacobian‑determinant criterion to guarantee functional independence of the integrals, ensuring that each constructed map is locally invertible.

The second part of the paper addresses a system of two coupled second‑order ODEs. Each equation supplies its own first integral, (I_1) and (I_2). By crossing these integrals—using (I_1) as the new independent variable for the first equation and (I_2) as the new dependent variable for the second, and vice versa—the authors obtain “cross‑integral transformations.” These lead to a linear coupled second‑order system, or, after an additional non‑local step, to a completely decoupled linear first‑order system. The paper illustrates the procedure with several well‑known nonlinear models (e.g., the Riccati pair, coupled Duffing‑type oscillators), each yielding four or more distinct linearizations.

Throughout the manuscript the authors accompany the theoretical development with explicit examples, showing step‑by‑step how the integrals are derived, how the new variables are defined, and how the transformed linear equations are solved. They verify that the solutions of the linearized equations, when pulled back through the inverse transformation, reproduce the original nonlinear solutions, thereby confirming the correctness of each map.

In conclusion, the authors claim that their algorithm is simple, systematic, and computationally efficient. It not only recovers all known linearizing transformations (point, contact, Sundman, generalized Sundman, etc.) but also uncovers previously unknown families, such as the direct integral and multi‑integral chain transformations. The method promises broad applicability: it can be automated within symbolic computation environments, extended to higher‑dimensional systems, and used as a preprocessing step for numerical integration or control‑design tasks where linear models are preferable. Future work is suggested on (i) extending the approach to partial differential equations, (ii) exploring geometric interpretations of the new transformations, and (iii) integrating the algorithm into software packages for automated symmetry and integrability analysis.