Strict disintegratedness of generic Painleve equations

We study generic equations in the Painleve families I-VI, and prove (at least for the families I-V) that if $y_{1},\ \ldots,\ y_{n}$ are distinct solutions, then $y_{1},\ y_{1}',\ \ldots,\ y_{n},\ y_{n}'$ are algebraically independent over $\mathbb{C}(t)$. For the single equation $P_{I}$ : $y'=6y^{2}+t$ this was already proved by Nishioka. Our methods involve both elementary and advanced notions from the model theory of differentially closed fields, together with information on the global structure of the Painleve families (``irreducibility'', existence and nature of algebraic solutions) obtained by the Japanese school.