Computational Methods and Function Theory

, Volume 3, Issue 1, pp 117–126 | Cite as

Airy Solutions of Painlevé’s Second Equation

  • Norbert Steinmetz


We prove that any transcendental solution of Painlevé’s second equation w″ = α + zw + 2w 3, which has the form w = R(z,u), with R rational in both variables and non-linear with respect to u, is obtained by repeated application of the Bäcklund transformation to some solution of the Riccati equation U′ = ±(z/2 + U 2). In particular, \(\alpha = n+1/2, n\in {\rm Z}\), and w has order of growth ϱ 3/2. Moreover it is shown that u satisfies some Riccati differential equation u’ = a(z) + b(z)u + c(z)u with rational coefficients.


Painlevé Airy and Riccati differential equation Bäcklund transformation 

2000 MSC

34M55 30D35 


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Copyright information

© Heldermann  Verlag 2003

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität DortmundDortmundGermany

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