Fermat and Malmquist type matrix differential equations

Abstract

The systems of nonlinear differential equations of certain types can be simplified to matrix forms. Two types of matrix differential equations will be considered in the paper, one is Fermat type matrix differential equation

$$A{(z)^n} + A'{(z)^n} = E$$

where n = 2 and n = 3, another is Malmquist type matrix differential equation

$$A'(z) = \alpha A{(z)^2} + \beta A(z) + \gamma E,$$

, where α (≠ 0), β, γ are constants. By solving the systems of nonlinear differential equations, we obtain some properties on the meromorphic matrix solutions of the above matrix differential equations. In addition, we also consider two types of nonlinear differential equations, one of them is called Bi-Fermat differential equation.