Example

In this section we consider an example where some classical results cannot be applied. However, we can apply the Theorem 3.6. Indeed, consider the differential equation

$$y^{(\mathrm{ iv})}-5y''+[\sin(t^q)+4]y=0, \mbox{with $q\in]2,\infty[$,}$$ (12)

which is of type (1.1) with $(a_0,a_1,a_2,a_3)=(4,0,-5,0)$ and $(r_0,r_1,r_2,r_3)(t)=(\sin(t^q),0,0,0).$ Then, the classical generalizations of Poincaré type theorems [26], the Levinson Theorem [9, Theorem 1.3.1], the Hartman-Wintner [9, Theorem 1.5.1] or the Eastham Theorem [9, Theorem 1.6.1] can not be applied to obtain the asymptotic behavior of (5.1). However, the hypotheses of Theorem 3.6 are satisfied see [7]. Then, the asymptotic formulas are given by

$$y_1(t) = e^{2t}\exp\Big(\frac{1}{12}\int_{t_0}^t \Big\{\sin(s^q)- f_1(s)\Big\}ds\Big),$$

$$y_2(t) = e^{t}\exp\Big(-\frac{1}{6}\int_{t_0}^t \Big\{\sin(s^q)- f_2(s)\Big\}ds\Big),$$

$$y_3(t) = e^{-t}\exp\Big(\frac{1}{6}\int_{t_0}^t \Big\{\sin(s^q)- f_3(s)\Big\}ds\Big),$$

$$y_4(t) = e^{-2t}\exp\Big(-\frac{1}{12}\int_{t_0}^t \Big\{\sin(s^q)- f_4(s)\Big\}ds\Big),$$

where

$$f_1(t) = 3(z'_1(s))^2+24z''_1(s)+4z_1(s)z''_1(s) +6[z_1(s)]^2z'_1(s) +8[z_1(s)]^3+[z_1(s)]^4$$

$$f_2(t) = 3(z'_2(s))^2+12z_2^2(s)+4z_2(s)z''_2(s) +12[z_2(s)]^2z'_2(s) +4[z_2(s)]^3+[z_2(s)]^4$$

$$f_3(t) = 3(z'_3(s))^2-6z_3^2(s)+4z_3(s)z''_3(s) +6[z_3(s)]^2z'_3(s) -4[z_3(s)]^3+[z_3(s)]^4$$

$$f_4(t) = 3(z'_4(s))^2-12z_4^2(s)+4z_4(s)z''_4(s) +6[z_4(s)]^2z'_4(s) -8[z_4(s)]^3+[z_4(s)]^4$$

and $z_i(t)$ satisfies the following asymptotic behavior

$$\begin{align*}z_{i}(t),z'_{i}(t),z''_{i}(t) = \left\{ \begin{array}{lll} \displa... ...(s^p)\vert ds\Big), & & i=4,\quad \beta\in ]0,1]. \end{array}\right.\end{align*}$$