On the growth of solutions of certain second-order nonlinear differential equations

In this paper, we determine the growth of real-valued solutions of certain second-order algebraic differential equations. Our main result, together with a result of G. Valiron, shows that if y₀ is an entire function which has only real, nonnegative coefficients in its power series around the origin, and which is a solution of a quadratic second-order algebraic differential equation, then y₀ satisfies a growth estimate of the form,y₀(x) ? exp(exp x^c), where c is a constant, for all sufficiently large x. The determination of the growth of such solutions was an open problem since the Valiron-Wiman theory fails to provide any information on growth, if the equation possesses a solution of infinite order of growth.