Defintion of Differential Equations

 

Partial Differential Equation

Ordinary Differential Equation

Order of Differential Equation

Grade of Differential Equation

Linear and Homogeneous Differential Equation

 

Introduction

Differential is called an equation that involves an unknown function with its derivatives. Depending on the number of the independent variables we have partial and ordinary differential equations.

 

Partial Differential Equation

This is the general case of a differential equation that the unknown function depends on many independent variables. Generally a partial differential equation is of the form:

Where n and m are real numbers.

The equation includes the different independent variables, the unknown function and the partial derivatives of this unknown function with respect to the different independent variables. Many of these factors in a real differential equation may be missing. For example a partial differential equation that depends on two independent variables x and t is:

 

Ordinary Differential Equation

If the differential equation depends only on one independent variable, it is called ordinary differential equation. Generally an ordinary differential equation is of the form:

Where n is real number.

An example of a differential equation of this kind is:

 

 

Order of Differential Equation

The order of the differential equation is defined by the highest derivative it contains. For example:

 

 

Grade of Differential Equation

The grade of the differential equation is defined by the highest power of the highest grade of the derivatives it contains. For example:

 

 

Linear and Homogeneous Differential Equation

Linear is called the first grade differential equation of the form:

Where

Are only functions of x.

The differential function:

is called the homogeneous of the corresponding linear differential equation.

 An example of linear differential equation is:

And its corresponding homogeneous is: