We show that the following nonlinear boundary-value problem has a positive solution: \[ \ddot{y}-y+yF(y^2,t)=0,y(0)=0=\lim_{t\to \infty}\,y(t),\dot{y}(0) \mathrm{ finite.} \] We assume that \(F\) is continuous for \(t>0\), \(y^2 \geq 0\); \(F>0\) for \(y^2 >0\); and \(\eta^{-\delta}F(\eta ,t)\) is strictly increasing in \(\eta\) for some \(\delta>0\). In addition, we assume that for all real \(c\), \[ \lim_{t\to \infty} \, F(c^2,t)=0,\mathrm{ and }\int_{0}^{1}\, t^{1-\epsilon}F(c^2t,t)dt 0\). This last assumption allows \(F\) to be singular at \(t = 0\). Our technique is based on that of \textit{Z.Nehari} [Proc. roy. Trish Acad., Sect. A 62, 117-135 (1963; Zbl 0124.30204)] and \textit{G. H. Ryder} [Pacific J. Math., 22, 477-503 (1967; Zbl 0152.28303)], and extends their results. Our results also extend some of those of \textit{G. Sansone} [Sympos. Math., Roma 6, Meccanica non lineare Stabilità 1970, 3-139 (1971; Zbl 0249.34038)] to more general nonlinearities.