NCERT Class 12-Mathematics: Chapter –9 Differential Equations Part 1

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9.1 Overview

(i) An equation involving derivative (derivatives) of the dependent variable with respect to independent variable (variables) is called a differential equation.

(ii) A differential equation involving derivatives of the dependent variable with respect to only one independent variable is called an ordinary differential equation and a differential equation involving derivatives with respect to more than one independent variables is called a partial differential equation.

(iii) Order of a differential equation is the order of the highest order derivative occurring in the differential equation.

(iv) Degree of a differential equation is defined if it is a polynomial equation in its derivatives.

(v) Degree (when defined) of a differential equation is the highest power (positive integer only) of the highest order derivative in it.

(vi) A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.

(vii) To form a differential equation from a given function, we differentiate the function successively as many times as the number of arbitrary constants in the given function and then eliminate the arbitrary constants.

(viii) The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.

(ix) ‘Variable separable method’ is used to solve such an equation in which variables can be separated completely, i.e., terms containing x should remain with dx and terms containing y should remain with .

(x) A function is said to be a homogeneous function of degree if for some non-zero constant .

(xi) A differential equation which can be expressed in the form where and are homogeneous functions of degree zero, is called a homogeneous differential equation.

(xii) To solve a homogeneous differential equation of the type , we make substitution and to solve a homogeneous differential equation of the type , we make substitution .

(xiii) A differential equation of the form, where P and Q are constants or functions of only is known as a first order linear differential equation. Solution of such a differential equation is given by, where .

(xiv) Another form of first order linear differential equation is , where and are constants or functions of only. Solution of such a differential equation is given by , where .

9.2 Solved Examples

Short Answer (S.A)

Question 1:

Find the differential equation of the family of curves .

Answer:

Given

Differentiating on both sides with respect to

Question 2:

Find the general solution of the differential equation

Answer:

Question 3:

Given that and . Find the value of when .

Answer:

Given that

Substituting and , we get

Therefore, .

Now, substituting in the above, we get .

Question 4:

Solve the differential equation

Answer:

The equation is of the type , which is a linear differential equation.

Now

Therefore, solution of the given differential equation is

Hence

Question 5:

Find the differential equation of the family of lines through the origin.

Answer:

Let be the family of lines through origin. Therefore,

Eliminating , we get

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