**2.1 First case: Δ > 0**

For Δ >0, the solution of the i.v.p. (4)-(5) is given by

$$\mathbf{x}(t) = \sqrt{\frac{\sqrt{\Delta} - p}{q}} \mathbf{cn} \left( \sqrt[4]{\Delta} t - \text{sign}(\dot{\mathbf{x}}\_0) \mathbf{cn}^{-1} \left( \sqrt{\frac{q}{\sqrt{\Delta} - p}} \mathbf{x}\_0, \frac{1}{2} - \frac{p}{2\sqrt{\Delta}} \right), \frac{1}{2} - \frac{p}{2\sqrt{\Delta}} \right). \tag{10}$$

Making use of the additional formula

$$\text{cn}(\mathbf{x} + \mathbf{y}, m) = \frac{\text{cn}(\mathbf{x}, m)\text{cn}(\mathbf{y}, m) + \text{sn}(\mathbf{x}, m)\text{dn}(\mathbf{x}, m)\text{sn}(\mathbf{y}, m)\text{dn}(\mathbf{y}, m)}{\mathbf{1} - m\text{sn}(\mathbf{x}, m)\text{sn}(\mathbf{y}, m)},\tag{11}$$

the solution (10) could be expressed as

$$\mathbf{x}(t) = \frac{\mathbf{x}\_0 \mathbf{cn}(\sqrt{a}t|m) + \frac{\dot{\mathbf{x}}\_0}{\sqrt{a}} \mathbf{dn}(\sqrt{a}t|m) \mathbf{sn}(\sqrt{a}t|m)}{\mathbf{1} + \frac{p + q \mathbf{x}\_0^2 - a}{2\sqrt{\Delta}} \mathbf{sn}\left(\sqrt[4]{\Delta}t|m\right)^2},\tag{12}$$

where

$$m = \frac{1}{2} \left( 1 + \frac{p}{\sqrt{\Delta}} \right) \text{ and } \boldsymbol{\omega} = \sqrt[4]{\Delta}. \tag{13}$$

Solution (12) is a periodic solution with period

$$T = 4 \left| \frac{K(m)}{\sqrt{\alpha}} \right|. \tag{14}$$

#### **Example 1.**

Let us consider the i.v.p.

$$\begin{cases} \varkappa''(t) + \varkappa(t) + \varkappa^3(t) = \mathbf{0}, \\\varkappa(\mathbf{0}) = \mathbf{1} \,\&\,\varkappa'(\mathbf{0}) = -\mathbf{1}. \end{cases} \tag{15}$$

Using formula (10), the exact solution of the i.v.p. (15) reads

$$\mathbf{x}(t) = -\sqrt{\sqrt{6}-1}\mathbf{cn}\left(\sqrt[4]{6}t + \mathbf{cn}^{-1}\left(-\frac{1}{\sqrt{\sqrt{6}-1}}, \frac{1}{2} - \frac{1}{2\sqrt{6}}\right), \frac{1}{12}\left(6 - \sqrt{6}\right)\right). \tag{16}$$

*Analytical Solutions of Some Strong Nonlinear Oscillators DOI: http://dx.doi.org/10.5772/intechopen.97677*

According to the relation (12)-(13), the exact solution of the i.v.p. (15) is also written as

$$\mathbf{x}(t) = \frac{2\sqrt[4]{6}\text{dn}\left(\sqrt[4]{6}t|\frac{1}{12}\left(\mathbf{6}-\sqrt{\mathbf{6}}\right)\right)\text{sn}\left(\sqrt[4]{6}t|\frac{1}{12}\left(\mathbf{6}-\sqrt{\mathbf{6}}\right)\right) - 2\sqrt{\mathbf{6}}\text{cn}\left(\sqrt[4]{6}t|\frac{1}{12}\left(\mathbf{6}-\sqrt{\mathbf{6}}\right)\right)}{\left(\sqrt{\mathbf{6}}-2\right)\text{sn}\left(\sqrt[4]{6}t|\frac{1}{12}\left(\mathbf{6}-\sqrt{\mathbf{6}}\right)\right)^{2} - 2\sqrt{\mathbf{6}}},\tag{17}$$

and its periodicity is given by

$$T = \frac{4K\left(\frac{1}{12}\left(6 - \sqrt{6}\right)\right)}{\sqrt[4]{6}} \approx 3.27458.4$$

In **Figure 1**, the comparison between the exact analytical solution (17) and the approximate numerical RK4 solution is presented. Full compatibility between the two analytical and numerical solutions is observed.
