Kokanduni uz

 
Keywords:
First-order differential equations, Linear differential equations, Initial value 
problems, Exact equations, Bernoulli equations 
Introduction: 
Differential equations are mathematical equations that describe how a 
variable change with respect to another variable. In many fields of science and engineering, 
differential equations play a crucial role in modeling and predicting physical phenomena. 
First-order linear differential equations are a type of differential equation that is frequently 
encountered in applications. Bernoulli's differential equation is a type of nonlinear differential 
equation that can be transformed into a linear equation. In this paper, we will provide an 
overview of first-order linear differential equations and Bernoulli's differential equation, as 
well as the methods used to solve them. 
Literature review: 
Many authors have investigated the properties and solutions of first-
order linear differential equations. In their book "Differential Equations with Applications and 
Historical Notes," George F. Simmons and Steven G. Krantz provide a comprehensive 
introduction to differential equations and their applications [3]. They discuss the general form 
of first-order linear differential equations, as well as methods for solving them using 
integrating factors. Also, a couple of examples have been provided for applications of first-
order linear differential equations in various fields, such as physics and biology [4]. Several 
authors have also studied the properties and solutions of Bernoulli's differential equation [5]. 
In their paper "Bernoulli's Differential Equation Revisited," John A. Pelesko and David H. 
Bernstein investigate the properties of Bernoulli's differential equation and its solutions [2]. 
They provide a detailed derivation of the transformation that turns Bernoulli's differential 
equation into a linear equation, and they discuss the properties of the solutions of the 
transformed equation. They also provide several examples of applications of Bernoulli's 
differential equation in physics and engineering. In his book "Differential Equations and 
Linear Algebra," C. Henry Edwards discusses the general theory of differential equations and 
their applications [1]. He provides a detailed discussion of first-order linear differential 
equations and the method of integrating factors, as well as the properties and solutions of 
Bernoulli's differential equation. He also discusses the use of differential equations in 
modeling real-world problems, such as population growth and chemical reactions.