Differential Equations, Objectives, Differential Equations, Order and Degree of a Differential Equations, Linear and Non-Linear Differential Equation, Formation of a Differential Equation

Glide to success with Doorsteptutor material for IAS : Get complete video lectures from top expert with unlimited validity: cover entire syllabus, expected topics, in full detail- anytime and anywhere & ask your doubts to top experts.

Download PDF of This Page (Size: 164K)

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.

Order and Degree of a Differential Equations
Order and Degree of a Differential Equations

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.

All straight lines passing through the origin

All Straight Lines Passing through the Origin

All straight lines passing through the origin

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.

Developed by: