All admissible meromorphic solutions of certain type of second degree second order algebraic ODEs

Following the same idea of Halburd and Wang [9] to construct small functions of w and w^′ by using the first one or two terms in the local series expansion for w at zeros, we solve all the admissible meromorphic solutions of the second order algebraic differential equation w^″w−w^′2+aww^′+bw²=αw+βw^′+γ, where a,b are constants and α,β,γ are small meromorphic functions of w in the sense of Nevanlinna theory. These solutions can have infinite order as Hayman [11] has pointed out but still holds for his conjecture that T(r,w)≤c₁e^c₂rc, 0≤r<+∞ when α,β,γ are rational functions, where c₁,c₂ and c are some positive constants.