Integrability cases for the anharmonic oscillator equation

TL;DR

This paper explores integrability conditions for the nonlinear anharmonic oscillator equation, deriving explicit solutions through transformations and Riccati and Bernoulli equations, expanding understanding of its solvable cases.

Contribution

The paper introduces new classes of exact solutions for the anharmonic oscillator by applying Euler's theorem and transforming integrability conditions into Riccati and Bernoulli equations.

Findings

Derived three classes of solutions for the nonlinear oscillator.

Established integrability conditions as Riccati equations.

Reduced conditions to Bernoulli equations for explicit solutions.

Abstract

Using N. Euler's theorem on the integrability of the general anharmonic oscillator equation \cite{12}, we present three distinct classes of general solutions of the highly nonlinear second order ordinary differential equation $\frac{d^{2}x}{dt^{2}}+f_{1}\left(t\right) \frac{dx}{dt}+f_{2}\left(t\right) x+f_{3}\left(t\right) x^{n}=0$. The first exact solution is obtained from a particular solution of the point transformed equation $d^{2}X/dT^{2}+X^{n}\left(T\right) =0$, $n\notin \left\{-3,-1,0,1\right\} $, which is equivalent to the anharmonic oscillator equation if the coefficients $f_{i}(t)$, $i=1,2,3$ satisfy an integrability condition. The integrability condition can be formulated as a Riccati equation for $f_{1}(t)$ and $\frac{1}{f_{3}(t)}\frac{df_{3}}{dt}$ respectively. By reducing the integrability condition to a Bernoulli type equation, two exact classes of solutions of the anharmonic oscillator equation are obtained.