A note on deriving linearizing transformations for a class of second order nonlinear ordinary differential equations

We present a method of deriving linearizing transformations for a class of second order nonlinear ordinary differential equations. We construct a general form of a nonlinear ordinary differential equation that admits Bernoulli equation as its first integral. We extract conditions for this integral to yield three different linearizing transformations, namely point, Sundman and generalized linearizing transformations. The explicit forms of these linearizing transformations are given. The exact forms and the general solution of the nonlinear ODE for these three linearizables cases are also enumerated. We illustrate the procedure with three different examples.

💡 Research Summary

The paper introduces a systematic method for linearizing a broad class of second‑order nonlinear ordinary differential equations (ODEs) by exploiting the existence of a first integral that can be cast into a Bernoulli‑type equation. Starting from a generic rational first integral I(t,x,ẋ)=A(t,x)ẋ+B(t,x)·C(t,x)ẋ+D(t,x), the authors rewrite it as a first‑order Bernoulli equation ẋ = ã(t)x + b̃(t)x^q, where q is an integer and the coefficients are expressed through four time‑dependent functions r₁(t)…r₄(t) and an auxiliary function f(t). Substituting these forms back yields the explicit second‑order ODE coefficients a₂(t,x), a₁(t,x), a₀(t,x) (equations 9a‑c).

The central insight is to express the first integral as a ratio of two perfect derivatives: I = (dF/dt)/(dG/dt). Defining new dependent and independent variables w = F(t,x) and z = ∫G(t,x,ẋ)dt, the original nonlinear ODE is transformed into the free‑particle equation d²w/dz² = 0. Whether the transformation is a point transformation (PT), a Sundman transformation (ST), or a generalized linearizing transformation (GLT) depends on the structure of G:

• PT arises when G is itself a perfect derivative of a function of (t,x) only; this requires the compatibility condition r₁ = (ḟ/f)(q−1) + r₃ f (equation 17). Under this condition the transformation formulas (19) involve only t and x, and the general solution follows from integrating the Bernoulli equation, giving expression (20).

• ST is obtained when G does not contain ẋ and is not a perfect derivative, which the authors show can only happen if f(t)=0. In this case the first integral reduces to a polynomial in ẋ (equations 26‑28). The new independent variable becomes z = ∫(r₃ x^{1−q}+ r₄) dt, and the solution is expressed explicitly by (28).

• GLT corresponds to the situation where G includes ẋ but still forms a perfect derivative; this requires f(t) ≠ 0. The resulting transformation (31) has the independent variable z = ∫