Let $L\left(y\right)=y^{\left(n\right)}+q_{n-1}\left(t\right)y^{(n-1)}+\cdots +q_0\left(t\right)y,t\in [a,b)$, be an $n$-th order differential operator, $L^{*}$ be its adjoint and $p,w$ be positive functions. It is proved that the self-adjoint equation $L^{*}\left(p\left(t\right)L\left(y\right)\right)=w\left(t\right)y$ is nonoscillatory at $b$ if and only if the equation $L\left(w^{-1}\left(t\right)L^{*}\left(y\right)\right)=p^{-1}\left(t\right)y$ is nonoscillatory at $b$. Using this result a new necessary condition for property BD of the self-adjoint differential operators with middle terms is obtained.