NCERT Class 12-Mathematics: Chapter – 9 Differential Equations Part 5 (For CBSE, ICSE, IAS, NET, NRA 2023)

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Question 22: Fill in the blanks of the following:

(i) Order of the differential equation representing the family of parabolas is ________ .

Answer:

One; is the only arbitrary constant.

(ii) The degree of the differential equation is ________.

Answer:

Two; since the degree of the highest order derivative is two.

(iii) The number of arbitrary constants in a particular solution of the differential equation is ________ .

Answer:

Zero; any particular solution of a differential equation has no arbitrary constant.

(iv) is a homogeneous function of degree________ .

Answer:

Zero

(v) An appropriate substitution to solve the differential equation is________.

Answer:

(vi) Integrating factor of the differential equation is ________.

Answer:

given differential equation can be written as and therefore

(vii) The general solution of the differential equation is________.

Answer:

from given equation, we have

(viii) The general solution of the differential equation

Answer:

and the solution is .

(ix) The differential equation representing the family of curves is ________ .

Answer:

Differentiating the given function w. r. t. x successively, we get

And

is the differential equation.

(x) when written in the form , then ________.

Answer:

the given equation can be written as

This is a differential equation of the type

Question 23:

State whether the following statements are True or False.

(i) Order of the differential equation representing the family of ellipses having centre at origin and foci on x-axis is two.

Answer:

True, since the equation representing the given family is has two arbitrary constants.

(ii) Degree of the differential equation is not defined.

Answer:

True, because it is not a polynomial equation in its derivatives.

(iii) is a differential equation of the type but it can be solved using variable separable method also.

Answer: True

(iv) is not a homogeneous function.

Answer:

True, because .

(v) is a homogeneous function of degree .

Answer:

True, because .

(vi) Integrating factor of the differential equation is

Answer:

(vii) The general solution of the differential equation is .

Answer:

True, because given equation can be written as

(viii) The general solution of the differential equation is

Answer: False

(ix) is a solution of the differential equation

Answer:

True,

Which satisfies the given equation.

(x) is a particular solution of the differential equation

Answer: False

Because does not satisfy the given differential equation.