Differential Equations, Objectives, Differential Equations, Order and Degree of a Differential Equations, Linear and Non-Linear Differential Equation, Formation of a Differential Equation
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Objectives
After studying this lesson, you will be able to:
Define a differential equation, its order and degree;
Determine the order and degree of a differential equation;
Form differential equation from a given situation;
Illustrate the terms “general solution” and “particular solution” of a differential equation through examples;
Solve differential equations of the following types:
Find the particular solution of a given differential equation for given conditions.
Differential Equations
As stated in the introduction, many important problems in Physics, Biology and Social Sciences, when formulated in mathematical terms, lead to equations that involve derivatives.
Equations which involve one or more differential coefficients such as (or differentials) etc. and independent and dependent variables are called differential equations.
Order and Degree of a Differential Equations
Order: It is the order of the highest derivative occurring in the differential equation.
Degree: It is the degree of the highest order derivative in the differential equation.
Differential Equations | Order | Degree | |
1. | One | One | |
2. | One | Two | |
3. | Two | Two | |
4. | Three | One | |
5. | Four | Two |
Example:
Find the order and degree of the differential equation:
Solution:
The given differential equation is or
Hence order of the differential equation is and the degree of the differential equation is.
Linear and Non-Linear Differential Equation
A differential equation in which the dependent variable and all of its derivatives occur only in the first degree and are not multiplied together is called a linear differential equation.
A differential equation which is not linear is called non-linear differential equation.
For example, the differential equations
and are linear.
The differential equation is non-linear as degree of is two.
Further the differential equation is non-linear because the dependent variable and its derivative are multiplied together.
Formation of a Differential Equation
Consider the family of all straight lines passing through the origin (see Fig.).
This family of lines can be represented by .....
Differentiating both sides, we get .....
From and , we get …..
So and represent the same family.
Clearly equation is a differential equation.
Working Rule:
To form the differential equation corresponding to an equation involving two variables, say and and some arbitrary constants, say, etc.
Differentiate the equation as many times as the number of arbitrary constants in the equation.
Eliminate the arbitrary constants from these equations.
Example:
Form the differential equation representing the family of curves.
Solution:
……
Differentiating both sides, we get
……
Differentiating again, we get
……
From and , we get
or
Which is the required differential equation.