Defintion of Differential Equations
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.
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:
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:
The order of the differential equation is defined by the highest derivative it contains. For example:
The grade of the differential equation is defined by the highest power of the highest grade of the derivatives it contains. For example:
Linear is called the first grade differential equation of the form:
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: