A particle of mass is executing oscillations about the origin on the  X-axis. Its potential energy is  $V\left(x\right)=k|x{|}^3$, where  k is a positive constant. If the amplitude of oscillation is a , then its time period T  is

We know
$U\left(x\right)=k{\left|x\right|}^3$ 
since, $F=-\frac{\text{d}U\left(x\right)}{\text{d}x}=-3k|x{|}^2$  …(1)
$x=a\sin \left(\omega t\right)$ 
This equation always fits to the differential equation
$\frac{{\text{d}}^{2}x}{\text{d}{t}^2}=-{\omega }^{2}x$    or  $m\frac{{\text{d}}^{2}x}{\text{d}t}=-m{\omega }^{2}x$
$\Rightarrow F=-m{\omega }^{2}x$ …(2)
Equation (1) and (2) give $-3k|x{|}^2=-m{\omega }^{2}x$ 
$$\begin{gathered} \Rightarrow \omega =\sqrt{\frac{3kx}{m}}=\sqrt{\frac{3ka}{m}\left[\sin \left(\omega t\right)\right]} \\ \Rightarrow \omega \propto \sqrt{a} (\because \omega =\frac{2\text{π}}{T}) \\ \Rightarrow T\propto \frac{1}{\sqrt{a}} \end{gathered}$$