Differential Equations

In this page differential equations we are going to see different topics in the major topic differential equation.

Definition:

An equation involving one dependent variable and its derivatives with respect to one or more independent variables is called a Differential Equation.

If y = f(x) is a given function,then its derivative dy/dx is known as rate of change of y with respect to x. In any natural process the variables involved and their rates of change are connected with one another by means of the basic scientific principles that govern the process. When this expression is written in mathematical symbols,the result is often a differential equation. There are two types of differential equation

(i) Ordinary

(ii) Partial

Ordinary differential equation

Partial differential equation

An ordinary differential equation is a differential equation in which a single independent variable enters wither explicitly or implicitly.

A partial differential equation is one which atleast two independent variables occur and the partial differential coefficients occurring in them have reference to any of these variable.

Examples:

(i) dy/dx = x + 5

(ii) (d²y/dx²) - 4(dy/dx) + 3 y = 0 are the examples of ordinary differential equation.

Example:

x (∂z/∂x) + y (∂z/∂y) = 2

Order and degree of a differential equation:

The "order" of a differential equation is the order of the highest order derivative occurring in it. The "degree" of the differential equation is the degree of the highest order derivative which occurs in it.

Now let us see some examples to know how to find order and degree of any differential equation.

Example 1:

d³y/dx³ + (d²y/dx²)³ + (dy/dx)⁵ + y = 7

here "y" is differentiated three times with respect to "x". So the order is 3 and its degree is 1.

Example 2:

(d²y/dx²) = [4 + (dy/dx)⁵ + y = 7

here "y" is differentiated three times with respect to "x". So the order is 3 and its degree is 1.

differential equations differential equations