Solutions of quaternion-valued differential equations with or without commutativity

Most results on quaternion-valued differential equation (QDE) are based on J. Campos and J. Mawhin's fundamental solution of exponential form for the homogeneous linear equation, but their result requires a commutativity property. In this paper we discuss with two problems: What quaternion function satisfies the commutativity property? Without the commutativity property, what can we do for the homogeneous equation? We prove that the commutativity property actually requires quaternionic functions to be complex-like functions. Without the commutativity property, we reduce the initial value problem of the homogeneous equation to a real nonautonomous nonlinear differential equation.