A modification to the first integral method and its applications

Abstract

In this article, we modify Feng’s first integral method (FIM) for the purpose of enlarging its applications. Compared with original FIM, our modified FIM is more straight and can also be employed to find first integral of higher-order ordinary differential equations (ODEs). We employ our modified FIM into five differential equations, namely, the density-dependent conformable fractional diffusion-reaction equation, the Duffing-van der Pol oscillator, the complex cubic-quintic Ginzburg–Landau equation, the well-known nonlinear evolution equation for description of surface waves in a convecting liquid, and the KdV–Burgers–Fisher equation. Consequently, we get the same first integral obtained by Feng’s FIM for the first equation; for the second equation, we reobtain certain important first integral reported previously; for the third equation, we construct a new first integral of complex cubic-quintic Ginzburg–Landau equation; and for the fourth and fifth equations, we show the effectiveness of our approach to third-order ODEs and reobtain the same first integral recently presented by Kudryashov for the fourth equation, and for the fifth equation, two new first integrals are presented. All the above fully reveal the effectiveness of our modification.