On the growth of transcendental entire solutions of algebraic differential equations

摘要

In this paper, we investigate the growth of transcendental entire solutions of the following algebraic differential equation a(z)f'~2 + (b_2(z)f~2 + b_1 (z)f + b_0 (z))f' = d_3(z)f ~3+ d_2(z)f~2 + d_1(z)f + d_0(z), where a(z), b_i(z) (0≤i≤2) and d_j(z) (0≤j≤3) are all polynomials, and this equation relates closely to the following well-known algebraic differential equation C(z,w)w'~2 + B(z, w)w' + A(z, w) = 0, where C(z, w) not ident to 0, B(z, w) and A(z, w) are three polynomials in z and w. We give relationships between the growth of entire solutions and the degrees of the above three polynomials in detail.