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First Order Differential Equations
Book chapter

First Order Differential Equations

P. Mohana Shankar
Differential Equations, pp 5-80
2018
DOI: https://doi.org/10.1201/b22396-2

Abstract

Runge Kutta Methods Real Roots Autonomous Systems Simple Analytical Solution Order Linear Differential Equation Equilibrium Study Equilibrium Point Separable Functions Unstable Solution Unknown Constant Higher Order Differential Equations Separable Differential Equation Analytical Solution Symbolic Toolbox Row Reduced Echelon Forms Differential Equation Implicit Solution MATLAB Script Eom Complex Pair Gompertz’s Equation Explicit Solution Integrating Factors Autonomous Differential Equation Order Differential Equation
First order differential equations have been discussed in this chapter. Differential equations may be classified as linear, nonlinear, or autonomous. Solutions can be found using the method based on integrating factors or the method on the separation of functions. Solutions may be obtained using the dsolve(.) command in Matlab for numerical approaches based on Runge-Kutta methods (when initial conditions are available). Even when analytical solutions are available, dsolve(.) and the Runge- Kutta methods can provide means for the verification of the results. Particular attention is paid to obtaining exact solutions using the method of integrating factors when the first order differential equation is linear (depends only on the independent variable y) and use of the separable function approach when the first order differential equation is linear or nonlinear. The D-field patterns which can provide insight into the behavior of the system described by the first order differential equation, are also explored. The analysis of autonomous systems is carried out in detail. The method of separable functions is used to obtain implicit solutions (in the absence of exact solutions, use of Runge-Kutta methods) and multiple ways of understanding the three forms of equilibria associated with these systems are provided. Plenty of descriptive and self-contained examples are given providing a broader view of the topics covered through multiple ways of verification of the results. Most physical and chemical phenomena involve changes taking place with respect to one or more parameters. This means that in its simplest form, systems responsible for these phenomena can be modeled in terms of an independent parameter and a dependent parameter. Differential equations of first order are characterized by the existence of the derivative of the first order in an equation containing an independent variable and a dependent variable. First order linear differential equations can be solved using the concept of integrating factors. While the method of integrating factors is applicable only to linear first order differential equations, linear and non-linear first order differential equations may be solved using the method of separable differential equations. Differential equations may be classified as linear, nonlinear, or autonomous. Solutions can be found using the method based on integrating factors or the method on the separation of functions.

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