12 class Maths Notes Chapter 9 Differential Equations  free PDF| Quick revision Differential Equations  Notes class 12 maths

CBSE Revision Notes for CBSE Class 12 Mathematics Differential Equations Definition, order and degree, general and particular solutions of a differential equation.Formation of differential equation whose general solution is given.Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree.

Class 12 Maths Chapter-9 Differential Equations  Quick Revision Notes Free Pdf

🔷 Chapter - 9 🔷
👉  Differential Equations  👈

1. INTRODUCTION

Differential equation constitute a very important part of mathematics as it has many applications in real life. Various laws of physics are often in the form of equations involving rate of change of one quantity with respect to another. As the mathematical equivalent of a rate is a derivative, differential equation arise very naturally in real life and methods for solving them acquire paramount importance.

Definition

An equation involving the dependent variable and independent variable and also the derivatives of the dependable variable is known as differential equation.

For example:

Differential equations which involve only one independent variable are called ordinary differential equation.

2. ORDER AND DEGREE OF DIFFERENTIAL EQUATIONS

2.1 Order

The order of a differential equation is the order of the highest derivative involved in the differential equation

For example:

2.2 Degree

The degree of a differential equation is the degree of the highest differential coefficient when the equation has been made rational and integral as far as the differential coefficients are concerned.

For example:

Illustration 1: Find the order and degree of the following differential equations.

3. FORMATION OF ORDINARY DIFFERENTIAL EQUATION

An ordinary differential equation is formed in an attempt to eliminate certain arbitrary constants from a relation in the variables and constants. Consider an equation containing n arbitrary constants. Differentiating this equation n times we get n additional equations containing n arbitrary constants and derivatives. Eliminating n arbitrary constants from the above (n + 1) equations, we obtain differential equation involving nth derivative.

Thus if an equation contains n arbitrary constants, the resulting differential equation obtained by eliminating these constants will be a differential equation of nth order.
i.e.,      an       equation     of          the            form

Illustration 2 : Find the differential equation of the family of all circles which pass through the origin and whose centre lie on y-axis

Sol. Let the equation of the circle be
x² + y² + 2gx + 2fy + c = 0
If it passes through (0, 0), then c = 0
•       The equation of circle is x² + y² + 2gx + 2fy = 0
Since the centre of the circle lies on y-axis then g =          0
•       The equation of the circle is
x² + y² + 2fy = 0.                                                 ...(i)
This represents family of circles.
Differentiating, we get

4. SOLUTION OF A DIFFERENTIAL EQUATION

The solution of the differential equation is a relation is a relation between the independent and dependent variable free from derivatives satisfying the given differential equation.

Thus the solution of dy/dx = m could be obtained by simply integrating both sides i.e., y = mx + c, where c is arbitrary constant.

(a) General solution (or complete premitive)

The general solution of a differential equation is the relation between the variables (not involving the derivatives) which contain the same number of the arbitrary constants as the order of the differential equation.

Thus the general solution of the differential equation

(b) Particular solution or Integral

A solution which is obtained by giving particular values to the arbitrary constants in the general solution is called a particular solution.

5. METHOD OF SOLVING AN EQUATION OF THE FIRST ORDER AND FIRST DEGREE

A differential equation of the first order and first degree can be written in the form

Method – 1

(i) Variable Separation:

The general form of such an equation is
f(x)dx + f(y)dy = 0                                             ..(i)
Integrating, we get
∫f(x )dx +∫f(y)dy = c which is the solution of (i)

(ii) Solution of differential equation of the type

Method -3

(i) Linear differential equation:

A differential equation is said to be linear if the dependent variable y and its derivative occur in the first degree.

where P and Q are functions of y only. To get the general solution of the above equations we shall determine a function R of x called Integrating function (I.F). We shall multiply both sides of the given equation by R

(ii) Differential equation reducible to the linear form:

Sometimes equations which are not linear can be reduced to the linear form by suitable transformation.

Method 4

Exact differential equation:

A differential equation is said to be exact if it can be derived from its solution (primitive) directly by differentiation, without any elimination, multiplication etc.

For example, the differential equation x dy + y dx = 0 is an exact differential equation as it is derived by direct differentiation for its solution, the function xy = c

Illustration 11 : Solve (1 + xy) y dx + (1 - xy) x dy = 0

Sol. The given equation can be written as
y dx + xy²dx + x dy x²y dy = 0
or,  (y dx + x dy) + xy (y dx - x dy) = 0
or,  d (xy) + xy (y dx - x dy) = 0

Dividing by x²y², we get
Which is the required solution.

Application of differential equations :

In solving some geometrical problems, the following results are very helpful.

Illustration 12: If the length of the sub-normal at any point P on the curve is directly proportional to OP², where O is the origin, then form the differential equation of the family of curves and hence find the family of curves.