We show that every solution of the first Painlevé equation has the finite growth order. The second and the fourth Painlevé equations have the same property.
Shun Shimomura. "The first, the second and the fourth Painlevé transcendents are of finite order." Proc. Japan Acad. Ser. A Math. Sci. 77 (3) 42 - 45, March 2001. https://doi.org/10.3792/pjaa.77.42