**Abstract**

Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing's oscillator. Using the exact analytical solution to cubic Duffing and cubicquinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form *x*€ þ*f x*ð Þ¼ 0 as well as *x*€ þ2*εx*\_ þ *f x*ð Þ¼ *F t*ð Þ, where *x* ¼ *x t*ð Þ and *f* ¼ *f x*ð Þ and *F t*ð Þ are continuous functions. In the present chapter, sometimes we will use *<sup>f</sup>*ð Þ¼� �*<sup>x</sup> f x*ð Þ and take the approximation *f x*ð Þ≈P*<sup>N</sup> <sup>j</sup>*¼<sup>1</sup>*<sup>p</sup> <sup>j</sup> x j* , where *j* ¼ 1, 3, 5, ⋯*N* only odd integer values and *x*∈ ½ � �*A*, *A* . Moreover, we will take the approximation *f x*ð Þ≈P*<sup>N</sup> <sup>j</sup>*¼<sup>0</sup>*<sup>p</sup> <sup>j</sup> x j* , where *j* ¼ 1, 2, 3, ⋯*N*, and *x*∈½ � �*A*, *A* . Arbitrary initial conditions are considered. The main idea is to approximate the function *f* ¼ *f x*ð Þ by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided.

**Keywords:** Nonlinear second-order equation, Duffing equation, Cubic-quintic Duffing equation, Helmholtz oscillator, Duffing-Helmholtz oscillator, Mixed parity oscillator, Damped Duffing equation, Damped Helmholtz equation, Forced Duffing equation, Nonlinear electrical circuit, Solitons
