This paper is concerned with the existence of bounded and $L^2-$solutions to equations of the form $$\stackrel{...}x+a(t)f(\dot{x})\ddot{x}+b(t)g(x)\dot{x}+c(t,x)= e(t),\leqno (*)$$ where $e(t)$ is a continuous square integrable function. We obtain sufficient conditions which guarantee that all solutions of the equation $(*)$ are bounded are in $L^{2}[0,\infty)$.