Abstract
In this paper we give a Lax formulation for a family of non-autonomous second-order differential equations of the type yzz+a3(z,y)yz3+a2(z,y)yz2+a1(z,y)yz+a0(z,y)=0. We obtain a sufficient condition for the existence of a Lax representation with a certain L-matrix. We demonstrate that equations with this Lax representation possess a quadratic rational first integral. We illustrate our construction with an example of the Rayleigh–Duffing–Van der Pol oscillator with quadratic damping.
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