In this paper we constructallPainlevé‐type differential equations of the form (d2y/dx2)2=F(x,y,dy/dx), whereFis rational inyandy′=dy/dx, locally analytic inx, and not a perfect square. No further simplifying assumptions are made, but it is found that the absence of a term linear iny″ in the class of equations under investigation forcesFto be a polynomial inyandy′. We find exactly six distinct classes of second‐degree Painlevé equations, denoted SD‐I,