The solutions of Painlevé’s Second Equation

Abstract. We prove that any transcendental solution of Painlevé’s second equation w′ ′ = α+ zw+ 2w3, which has the form w = R(z, u), 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 + U2). In particular α = n+ 1/2, n ∈ Z, and w has order of growth % = 3/2. Moreover it is shown that u satisfies some Riccati differential equation u ′ = a(z) + b(z)u+ c(z)u2 with rational coefficients