دراسة رياضية للحلول العامة للمعادلات التفاضلية الخطية من الرتبة الاولى

Abstract

The following scientific paper is a detailed mathematical analysis of the general solutions of the first order linear differential equation which belong to the basic classes of applied mathematics and are applied in modeling many physical and engineering processes. Different numerical techniques that can be used to get the solution of these equations with a better understanding of the dynamic system is the key interest of the study. It has also been spelled out that integrating factor method is efficient when handling equations with variable coefficients and hence qualifies to be applied say in circuitry electrical circuits. With reference to the other method called separation of variables, it is also easy to solve separable equations directly and is used in heat transfer, fluid flow among others. Also, Laplace analysis is highly flexible when finding dynamics in systems especially with complicated initial conditions and is therefore essential in engineering and automatic control.

The latter compares the three methods with detailed explanations of the valuable and the possible drawbacks of each of them as well as the possibilities of their application to solve practical tasks. Both methods were used on examples and results were discussed to demonstrate their efficiency and possible uses. The results presented are to highlight the

idea that in choosing the method for solution proper attention should be paid to the type of equation and the field of its possible application. For computations, the study suggests the use of mathematical methods in conjunction with present software to expedite and increase the efficiency of solutions. This paper makes an appeal of further research especially on the use of differential equations in the promotion of more efficient and flexible ways in the analysis of complex systems.