Linearizable Exact and Explicit Analytic Solutions for Riccati Differential Equations

In this paper, linearizable exact and explicit analytic solutions for classical Riccati differential equations are investigated in the complex field C by locally reducing the equations to other q-difference-differential equations and constructing invertible holomorphic solutions of the q-difference-differential equations. We discuss the existence of invertible analytic solutions of the q-difference-differential equations in three cases: (i) |q|≠1, (ii) |q|=1 but not a root of unity, and (iii) |q|=1 but a root of unity. Especially for case (ii), we deal with the q-difference-differential equations under the Brjuno condition, which is weaker than the Diophantine condition. Finally, we give an algorithm for solving the Riccati differential equations, which is helpful to obtain their approximate solutions.