In this paper, we first investigate the classification of positively homogeneous equations $(\phi_p(u'))'+q(t)\phi_p(u)=0$, $u(0)=0=u(1)$, where $p>1$ is fixed, $\phi_p(u)=|u|^{p-2}u$ and $q\in L^{\infty}(0,1)$, and then discuss the existence of solutions for non-homogeneous equations. The main method of classification is by using a generalized Prufer equation

$$ \theta'=|\4\cos_p\theta|^p+\frac{q(t)}{p-1}|\4\sin_p\theta|^p\quad\text{for }t\in(0,1), $$

where $\sin_p:\mathbb{R}\to[-1,1]$ is a periodic function and $\cos_pt=\mathrm{d}\sin_pt/\mathrm{d} t$ for $t\in\mathbb{R}$.