In this nice paper, the following differential (scalar) equation is considered: \[ u'=a(t)f(u)+b(t)g(u),\tag{1} \] where \(a:\mathbb{R}\to\mathbb{R}_+\) and \(b,f,g:\mathbb{R}_+\to\mathbb{R}_+=[0,\infty)\) are continuous functions satisfying the conditions: (i) \(f(u)>0\) for \(u>0\) and \(\int^\infty du/f(u)=\infty\), (ii) there exists a \(T\geq 0\) such that \(a(t)f(0)+b(t)g(0)\geq 0\) for \(t\geq T\). Under some extra assumptions sufficient conditions are derived for the global existence and for the boundedness of solutions of the equation (1) satisfying the initial condition \(u(t_0)= u_0\) \((t_0\geq T)\). Applications to the system \(x'=\phi(t,x)\) \((\phi:\mathbb{R}_+\times \mathbb{R}^n\to \mathbb{R}^n)\) are also deduced.