This paper is engaged with Painlevé’s differential equation P 34 :2 ww ″ = w ″ 2 + 2 w 2 (2 w − z ) − α , also known as Ince’s equation XXXIV and closely related to Painlevé’s second differential equation $${{\rm{P}}_\Pi}:\varpi\prime\prime= \alpha + z\varpi + 2{\varpi^3}$$ P Π : ϖ ″ = α + z ϖ + 2 ϖ 3 . We will show that the transcendental solutions belong to the Yosida class $${\mathfrak{Y}_{{\rm{1,}}{1 \over 2}}}$$ Y 1, 1 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 $${\mathfrak{Y}_{{1 \over 2}{\rm{,}}{1 \over 2}}}$$ Y 1 2 , 1 2 .

中文翻译：

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关于Painlevé的微分方程P34的注解

本文涉及Painlevé的微分方程P 34 :2 ww ″ = w ″ 2 + 2 w 2 (2 w − z ) − α，也称为Ince方程XXXIV，与Painlevé的二阶微分方程$${{\ rm{P}}_\Pi}:\varpi\prime\prime= \alpha + z\varpi + 2{\varpi^3}$$ P Π : ϖ ″ = α + z ϖ + 2 ϖ 3 . 我们将证明超越解属于 Yosida 类 $${\mathfrak{Y}_{{\rm{1,}}{1 \over 2}}}$$ Y 1, 1 2 并且没有缺陷理性目标。我们还将识别次正规解并证明它们的特征在于它们的第一个积分属于类 $${\mathfrak{Y}_{{1 \over 2}{\rm{,}}{ 1 \over 2}}}$$ Y 1 2 , 1 2 。