# NCERT Solutions for Class 12 Maths Maths Part-2 Chapter 9

## Differential Equations Class 12

### Exercise 9.1 : Solutions of Questions on Page Number : 382

Q1 :

Determine order and degree(if defined) of differential equation

The highest order derivative present in the differential equation is. Therefore, its order is four.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Q2 :

Determine order and degree(if defined) of differential equation

The given differential equation is:

The highest order derivative present in the differential equation is. Therefore, its order is one.

It is a polynomial equation in. The highest power raised tois 1. Hence, its degree is one.

Q3 :

Determine order and degree(if defined) of differential equation

Q4 :

Determine order and degree(if defined) of differential equation

Q5 :

Determine order and degree(if defined) of differential equation

Q6 :

Determine order and degree(if defined) of differential equation

Q7 :

Determine order and degree(if defined) of differential equation

Q8 :

Determine order and degree(if defined) of differential equation

Q9 :

Determine order and degree(if defined) of differential equation

Q10 :

Determine order and degree(if defined) of differential equation

Q11 :

The degree of the differential equation

is

(A) 3 (B) 2 (C) 1 (D) not defined

Q12 :

The order of the differential equation

is

(A) 2 (B) 1 (C) 0 (D) not defined

### Exercise 9.2 : Solutions of Questions on Page Number : 385

Q1 :

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Q5 :

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Q9 :

Q10 :

Q11 :

The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:

(A) 0 (B) 2 (C) 3 (D) 4

Q12 :

The numbers of arbitrary constants in the particular solution of a differential equation of third order are:

(A) 3 (B) 2 (C) 1 (D) 0

### Exercise 9.3 : Solutions of Questions on Page Number : 391

Q1 :

Q2 :

Q3 :

Q4 :

Q5 :

Q6 :

Form the differential equation of the family of circles touching the y-axis at the origin.

Q7 :

Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.

Q8 :

Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

Q9 :

Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

Q10 :

Form the differential equation of the family of circles having centre on y-axis and radius 3 units.

Q11 :

Which of the following differential equations hasas the general solution?

A.

B.

C.

D.

Q12 :

Which of the following differential equation hasas one of its particular solution?

A.

B.

C.

D.

### Exercise 9.4 : Solutions of Questions on Page Number : 395

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Q15 :

Find the equation of a curve passing through the point (0, 0) and whose differential equation is.

Q16 :

For the differential equation find the solution curve passing through the point (1, â€“1).

Q17 :

Find the equation of a curve passing through the point (0, â€“2) given that at any point on the curve, the product of the slope of its tangent and y-coordinate of the point is equal to the x-coordinate of the point.

Q18 :

At any point (x, y) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point (-4, -3). Find the equation of the curve given that it passes through (-2, 1).

Q19 :

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.

Q20 :

In a bank, principal increases continuously at the rate of r% per year. Find the value of r if Rs 100 doubles itself in 10 years (log­e 2 = 0.6931).

Q21 :

In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it worth after 10 years.

Q22 :

In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?

Q23 :

The general solution of the differential equation

A.

B.

C.

D.

### Exercise 9.5 : Solutions of Questions on Page Number : 406

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Q11 :

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Q16 :

A homogeneous differential equation of the form can be solved by making the substitution

A. y = vx

B. v = yx

C. x = vy

D. x = v

Q17 :

Which of the following is a homogeneous differential equation?

A.

B.

C.

D.

### Exercise 9.6 : Solutions of Questions on Page Number : 413

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Q15 :

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point (x, y) is equal to the sum of the coordinates of the point.

Q16 :

Find the equation of a curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

Q17 :

The integrating factor of the differential equation is

A. eâ€“x

B. eâ€“y

C.

D. x

Q18 :

The integrating factor of the differential equation.

is

A.

B.

C.

D.

### Exercise Miscellaneous : Solutions of Questions on Page Number : 419

Q1 :

For each of the differential equations given below, indicate its order and degree (if defined).

(i)

(ii)

(iii)

Q2 :

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

(i)

(ii)

(iii)

(iv)

Q3 :

Form the differential equation representing the family of curves given by where a is an arbitrary constant.

Q4 :

Prove that is the general solution of differential equation, where c is a parameter.

Q5 :

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

Q6 :

Find the general solution of the differential equation

Q7 :

Show that the general solution of the differential equation is given by (x + y + 1) = A (1 â€“ x â€“ y â€“ 2xy), where A is parameter

Q8 :

Find the equation of the curve passing through the point whose differential equation is,

Q9 :

Find the particular solution of the differential equation

, given that y = 1 when x = 0

Q10 :

Solve the differential equation

Q11 :

Find a particular solution of the differential equation, given that y = â€“ 1, when x = 0 (Hint: put x â€“ y = t)

Q12 :

Solve the differential equation

Q13 :

Find a particular solution of the differential equation , given that y = 0 when

Q14 :

Find a particular solution of the differential equation, given that y = 0 when x = 0

Q15 :

The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?

Q16 :

The general solution of the differential equation is

A. xy = C

B. x = Cy2

C. y = Cx

D. y = Cx2

Q17 :

The general solution of a differential equation of the type is

A.

B.

C.

D.

Q18 :

The general solution of the differential equation is

A. xey + x2 = C

B. xey + y2 = C

C. yex + x2 = C

D. yey+ x2 = C