Consider in a real Hilbert space $H$ the differential equation (inclusion) (E): $p(t)u''(t)+q(t)u'(t) \in Au(t) + f(t)$ a.e. in $(0, \infty)$, with the condition (B): $u(0) = x \in \overline{D(A)}$, where $A :D(A)\subset H\rightarrow H$ is a (possibly set-valued) maximal monotone operator whose range contains $0$; $p, q\in L^{\infty}(0,\infty )$, such that $\mathrm{ess} \inf \ p>0$, $\frac{q}{p}$ is differentiable a.e., and $\mathrm{ess} \inf , \big\[{(\frac{q}{p})}^2 + 2(\frac{q}{p})^{\prime}\big] >0$. We prove existence of a unique (weak or strong) solution $u$ to (E), (B), satisfying $a^{\frac{1}{2}}u \in L^{\infty}(0,\infty ;H)$ and $t^{\frac{1}{2}}a^{\frac{1}{2}}u^{\prime} \in L^2(0,\infty ;H)$, where $a(t)=\exp{\big( \int\_0^t \frac{q}{p}, d\tau \big) }$, showing in particular the behavior of $u$ as $t\rightarrow \infty$.