In this paper we present an approach toward the comprehensive analysis of the nonintegrability of differential equations in the form $\ddot{x}=f(x,t)$ which is analogous to Hamiltonian systems with $1+1/2$ degrees of freedom. In particular, we analyze the nonintegrability of some important families of differential equations such as Painlevé II, Sitnikov, and the Hill–Schrödinger equation. We emphasize Painlevé II, showing its nonintegrability through three different Hamiltonian systems, and also Sitnikov, in which two different versions including numerical results are shown. The main tool for studying the nonintegrability of these kinds of Hamiltonian systems is Morales–Ramis theory. This paper is a very slight improvement to the talk with a similar title delivered by the author at the SIAM Conference on Applications of Dynamical Systems in 2007.