= Solution
Rewrite the equation as
$$
y''+q(x)y=0,
\qquad
q(x)=\frac1{\sqrt{1+x^3}}.
$$
On $[2,6]$, one has $q(x)\leq1/3$. If a nontrivial solution had two zeros, choose two consecutive ones $\alpha<\beta$ in that interval. Then $\beta-\alpha\leq4<\pi\sqrt3$.
The solution
$$
\varphi(x)=
\cos\left(\frac{x-(\alpha+\beta)/2}{\sqrt3}\right)
$$
of $\varphi''+\frac13\varphi=0$ is strictly positive on $[\alpha,\beta]$. Since $q\leq1/3$ and the coefficients are not identical on this interval, the <Sturm comparison theorem> requires $\varphi$ to have a zero between $\alpha$ and $\beta$, a contradiction. Hence every nontrivial solution has
$$
\boxed{\text{at most one zero on }[2,6]}.
$$
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