First order differential equation with piecewise constant argument

Abstract

In recent years, differential equation with piecewise constant argument (DEPCA) have started to play crucial roles of the phenomena in the real world. There are many problems occurred involving piecewise constant arguments in the field of mathematics, science and engineering. However, behaviour of the system with piecewise constant arguments is more complex when compared to the corresponding continuous regular systems. This study focused on solving first order nonlinear differential equations with piecewise constant argument. There are two types of differential equation which are the first order differential equation and the Bernoulli-type equations. The mathematical approaches are based on the DEPCA of simple forms, especially in the form of [t] to which a continuous system is considered directly over a unit interval. In order to solve these equations, the condition of existence and uniqueness are used to determine both the types of equations for some class functions of g. Thus, an explicit form of solutions for a certain class of Bernoulli-type equations was solved indirectly by using the previous approached. At the same time, the periodicity conditions for these solutions were established. Several examples were explained for both types of differential equation and some graphs were obtained by using MATHEMATICA software.