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Linearizable Exact and Explicit Analytic Solutions for Riccati Differential Equations

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  • Linli Liu

    (Department of General Education, Shandong Xiehe University, Jinan 250109, China)

  • Jianguo Si

    (Department of General Education, Shandong Xiehe University, Jinan 250109, China
    School of Mathematics, Shandong University, Jinan 250100, China)

Abstract

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.

Suggested Citation

  • Linli Liu & Jianguo Si, 2026. "Linearizable Exact and Explicit Analytic Solutions for Riccati Differential Equations," Mathematics, MDPI, vol. 14(13), pages 1-14, June.
    • Handle: RePEc:gam:jmathe:v:14:y:2026:i:13:p:2302-:d:1978285
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