Solutions that oscillate more rapidly Which one of the following two equations has solutions that oscillate more rapidly?
$$x''+\sqrt{t^6+3t^5+1} x =0$$
$$ x''+2t^3 x =0$$
I was trying to use sturm separation theorem but not getting exact approach towards it!!
Is there any direct way to do it?
Edit: The book from where I have given this question has answer as first equation in the hints provided at the back of book.
Book name: &quot;A Textbook  on Ordinary  Differential  Equations&quot; by Shair Ahmad, Antonio Ambrosetti(Second Edition)-Chapter5-Exercise-D2
If we assume that $t$ is confined in an interval with small amplitude we may assume that $t^3$ is approximately constant. By recalling that a solution of $x''+ K x=0$ (with $K&gt;0$) has a frequency of $\frac{\sqrt{K}}{2\pi}$ we immediately get that a solution of $x''(t)+2t^3 x(t)=0$ oscillates faster than a solution of $x''(t)+\sqrt{t^6+3t^5+1}\,x(t)=0$.
Just solve them and plot the two:

The full forms have constants, that must be set for initial conditions.  For example $c_1 \cos t + c_2 \sin t$.  In the above, I've set all constants to $1$, just to reveal the periods... the focus of the question.