Differential Equations

Definitions

     Differential Equation: A differential equation is a relation between the independent, dependent variables and their differential coefficients.

Formation of a Differential Equation

     The equation y = mx has one arbitrary constant and its differential equation is of order 1. The equation y = mx + c has two arbitrary constants and its differential equation is of order 2.

Solution of a Differential Equation

     Solution of a Differential Equation: The functional relation-ship between the independent variable and the dependent variable (such as y = f(x)) which satisfies the given differential equation is called the solution of the differential equation.

Classification of Differential Equation

     Differential equation are classified according to their order and they are:
First Order Differential Equation, Higher Order Differential Equation, Linear Differential Equation, Non-linear differential equation.

An Alternative Form of a First-order First-degree Differential Equation

 

     The fact that, Dy - f(x, y) Dx ® 0 as Dx ® 0 is expressed by writing dy = f(x,y)dx is an alternative form of the differential equation

Some Methods of Solving First Order First Degree Differential Equation

     The different ways of solving differential equation are a follows: (a). Method of separation of variables, (b). Homogeneous differential equations, (c). Linear differential equations.

Special types of a Second Order Differential Equation

     Second-order differential equations, by definition, contain a second derivative, like d²y/dx², for example. As well as the second derivative, there may also be a first derivative in the equation and sometimes a term involving just y itself.

Applications

     1. A wet porous substance in the open air loses its moisture at the rate proportional to the moisture content. If a sheet hung in the wind loses half of its moisture during the first hour, when will it have lost?

Summary

     1. An equation involving independent variables dependent variable and their derivatives is called a differential equation.

     2. A differential equation which involves only one independent variable is called an ordinary differential equation.

Conclusion

     In this chapter we have the formation of differential equations and also some methods of solving the differential equations namely, variables separable methods, homogeneous differential equations and linear differential equations.