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)≤c1ec2rc, 0≤r<+∞ when α,β,γ are rational functions, where c1,c2 and c are some positive constants.