A Study of Movable Singularities in Non-Algebraic First-Order Autonomous Ordinary Differential Equations

Abstract

We studied the movable singularities of solutions of autonomous non-algebraic first-order ordinary differential equations in the form of y′=I(y(t)) and y′=I1(y(t))+I2(y(t))+⋯+In(y(t)), aiming to prove that all movable singularities of all complex solutions of these equations are at most algebraic branch points. This study explores the use of the constructing triangle method to analyze complex solutions of autonomous non-algebraic first-order ordinary differential equations. For complex solutions in the form of y=w+iv, we treat the constructing triangle method as a way to construct a right-angled triangle in the complex plane, with the lengths of the adjacent sides being w and v. We use the definitions of the trigonometric functions sin and cos (the ratio of the adjacent side to the hypotenuse) to represent the trigonometric functions of complex solutions y=w+iv. Since the movable singularities of the inverse functions of trigonometric functions are easy to analyze, the properties of the movable singularities of the complex solutions are then easy to deal with.