NCERT Class 12-Mathematics: Chapter – 9 Differential Equations Part 5 (For CBSE, ICSE, IAS, NET, NRA 2022)
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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.