Remarks on Painlevé’s differential equation P34

Abstract

This paper is engaged with Painleve’s differential equation P34:2ww″ = w″2 + 2w2(2w − z) − α, also known as Ince’s equation XXXIV and closely related to Painleve’s second differential equation $${{\rm{P}}_\Pi}:\varpi\prime\prime= \alpha + z\varpi + 2{\varpi^3}$$ . We will show that the transcendental solutions belong to the Yosida class $${\mathfrak{Y}_{{\rm{1,}}{1 \over 2}}}$$ and have no deficient rational targets. We will also identify the sub-normal solutions and prove that they are characterised by the fact that their first integrals belong to the class $${\mathfrak{Y}_{{1 \over 2}{\rm{,}}{1 \over 2}}}$$ .