A New Method for the Exact Solution of Duffing Equation

A lot of methods, such as Jacobian elliptic function analysis, are used to look for the explicit exact solution of Duffing differential equation. The key of the analysis is to construct quotient trigonometric function, and then nonlinear algebraic equation set theory and method are used for the solution of some kinds of nonlinear Duffing differential equation. In this paper, the exact solution of Duffing equation is obtained by using constant variation method, making use of the formula to solve cubic equations and general solution of the homogeneous equation of Duffing equation with appropriate Constant m and function ( ) f t .


Introduction
The Duffing equation is a non-linear second-order differential equation. The equation describes the motion of a damped oscillator with a more complicated potential than in simple harmonic motion; in physical terms, it models, for example, a spring pendulum whose spring's stiffness does not exactly obey Hooke's law. It is also an example of a dynamical system that exhibits chaotic behavior.
Many of the dynamic behavior of mechanical power model can be described by a single degree of freedom oscillator [1] [2] [3] [4] [5]. The systems can be described by the following equation , where a is an undetermined parameter, the In the last few years, many scientists work hard to look for exact solutions and to develop some efficient procedures. However, what the key of problem is, the different type of nonlinear differential equation can only be efficiently solved by different corresponding methods. For instant, the Jacobi elliptic function expansion method [7], the exp-function method [8], the F-expansion method [9], the tanh method [10], the auxiliary equation method [11], the simplest equation method [12], and linearized harmonic balance method [13] are reliable methods for obtaining exact solutions of specific types of nonlinear differential equations.
Keqiang Li (2013) studied the periodic orbits of the modified Duffing differential equation [14]. He paid his attention to the existence of multiple periodic solutions of asymptotically linear Duffing equations with resonance on the left side of the first eigenvalue using index theory and the three-critical-points theorem and obtained a new result. As the function in the right side of his equation is zero, how to look for the exact solution of the equation is a significative problem. Alex (2013) used Jacobi elliptic functions to derive the exact solution of the cubic-quintic Duffing oscillator [6]. He also proved that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind. Vasile (2011) has presented a method for obtaining exact solutions for Duffing Equations [15]. Marinca (2011) has presented a method for obtaining exact solutions for Duffing and double-well Duffing equations [16]. What he used is traditional quotient trigonometric function expansion method, and his solution procedure is similar to the exp-function method. However, his result in third section has a distance to reach perfect.
Exact analytical solutions for a family of autonomous ordinary differential equations arising in connection with many important applied problems such as a cubic-quintic Duffing oscillator, Helmholtz-Duffing oscillator, and nonlinear Schrödinger equation are considered. Recently, some theory and applications are found in many contexts, such as, Jiang et al. (2012) presented a new approach via the well-known Poincaré-Birkhoff theorem to obtain the existence of periodic solutions to impulsive problems, they considered an impulsive Duffing equation, and then found the possibility of applying a generalized form of the Poincaré-Birkhoff theorem due to Ding to construct infinitely many periodic solutions of the impulsive Duffing equation even in a resonance case [17]. Jaume and Ana (2015) studied the periodic solutions of a kind of non-autonomous Duffing differential equation with forced pendulum. And, they obtained the condition for that the non-autonomous Duffing differential system has a periodic solution such that it tends to the periodic solution when forced pendulum satisfied with some conditions [18]. Gholam and Emmanuel (2015) have provided a set of inequalities that helps in finding transformation to obtain solutions to other Duffing equations from a known set of solutions to a related equation in a systematic manner [19]. Alexander (2018) has obtained the sufficient and necessary conditions for the existence of a positive periodic solution to a kind of non-autonomous Duffing type equation [20]. David and Douglas (2018) perform a complete analysis of the effect on the asymptotic dynamics of the insertion of delays into a sort of unforced Duffing type equation [21].
One of the most exciting recent advances of nonlinear science has been the development of some methods to look for exact solutions of nonlinear differential equations. It is important that many mathematical models are described by nonlinear differential equations and exact solutions are preferable instead of approximate ones. The aim of this paper is to put forward constant variation method to derive the exact solution of a Duffing equation of the form (1.1). The exact solution of Duffing equation is obtained by solving cubic equations and looking for appropriate constant m and function ( ) f t .

Constant Variation Method
In general, the Duffing equation does not admit an exact symbolic solution.
However, many approximate methods work well:  Expansion in a Fourier series will provide an equation of motion to arbitrary precision.  The cubic term also called the Duffing term can be approximated as small and the system treated as a perturbed simple harmonic oscillator.  1), we obtain following differential

t e c t e c t e c t e c t e c t e
Because functions 1 ( ) c t and 2 ( ) c t are discretionary, hence, we let Then, Equation (2.4) becomes

t e e c t e e c t e e
Therefore, we obtain a more compact differential equation form for unknown function ( ) c t .
If we can find an undetermined constant m and an undetermined function ( ) f t , such that they satisfy with following equation  Its solutions can be expressed as Applying function (2.10) to equations set (2.7), we get 3 On the other hand, we can prove that 3 k is forever equal to 8k when 0 m = .
Actually, if 0 m = , then We have showed a few representative integer parameters m and functions in Table 1.
In fact, m is not only integer, but it maybe any fractional as well. Some values of m and respective function are showed in Table 2.

Applications
In this section, we shall give some examples to explain our constant variation method.

Example 1. Consider following Duffing equation
According to theorem 1, we can easily find a couple of m and ( ) f t (see Ta-ble 2), for instant, take m = 1/2 and     where β is a positive constant which may not be a small value.
We can easily obtain exact solution of Equation (3.4) in the same way as example 1.