In mathematics, an ordinary differential equation is an equation in which a function of only one independent variable and its derivatives play a role. The term "normal" is used in contrast to "partial differential equations". In partial differential equations, there are two or more variables.
Ordinary differential equations are divided into two categories: linear and nonlinear. The solutions of a typical linear differential equation can be summed with a fixed number or multiplied by a fixed number. These equations are fully and accurately identified and studied, and there are a variety of analytical answers for them. In contrast, there are ordinary nonlinear differential equations whose summability properties are not true for their answers. Solving these equations is generally more complex and can rarely be answered based on basic mathematical functions. Instead, for such equations, answers can be found either serially or in integral form. In addition, the answer to nonlinear differential equations can be estimated using graphical numerical methods, which can be implemented manually or by computer. These estimation methods can provide useful information in the absence of analytical and closed answers.
Differential equations have a wide range of applications in basic sciences such as mathematics, physics, chemistry, biology and astronomy, as well as engineering sciences such as mechanics, electricity, materials and chemical engineering. The differential equation is a set of mathematical equations that expresses the relationship between an unknown function of one or more independent variables and its derivatives of different orders relative to the independent variables. Many general laws of nature (in physics, chemistry, biology, and astronomy) find their most natural mathematical expression in the language of differential equations.
Given that most differential equations that are close to the reality of the phenomena have nonlinear and complex terms, it is difficult or impossible to find an analytical answer for them. The lack of analytical answers for such complex and nonlinear equations has led to the development of numerical solution methods. The most important evaluation parameters of numerical solution methods are speed, accuracy and correctness of the equation. Due to the dramatic increase in the speed and capacity of information processing in processors and computers in the late nineteenth century, numerical solution methods became more widespread, and this expansion continues.
Ordinary differential equations
Traces of ordinary differential equations can be found in various fields of mathematical, experimental, or social sciences, because these equations represent changes in mathematical language. Since in these equations functions, derivatives and differentials are interconnected, they can be used to express dynamic phenomena and change.
Examples of disciplines in which ordinary differential equations have a basic function are: mathematics such as geometry, engineering such as analytical mechanics and electrical engineering (electrical circuit behavior analysis), geology (water forecasting, and so on). Air), chemistry (analysis of nuclear reaction chains), biology (spread of infectious diseases, genetic changes), ecology and modeling of population and economy (changes in profits and stock prices)
Many prominent historian mathematicians have been involved in solving ordinary differential equations, including Newton, Leibniz, the Bernoulli family, Riccati, Alexei Chloro, D'Alembert, and Leonard Euler.