Solution
= Solution
Take
$$
q_1(x)=1,\qquad
\varphi_1(x)=\sin(x+1).
$$
Its consecutive zeros are $-1$ and $\pi-1$. For the stated equation,
$$
q_2(x)=2+\cos3x\geq1=q_1(x),
$$
and $q_2$ is not identically equal to $1$ on the interval. The <Sturm comparison theorem> therefore shows that every nontrivial solution has at least one zero in
$$
\boxed{[-1,\pi-1]}.
$$