# NCERT solutions Mathematics Textbook for Class 12 Part 2 chapter 3 Differential Equations

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 75- 180-

### Chapter 3 - Differential Equations

#### Pages 382 - 383

Determine order and degree(if defined) of differential equation (d^4y)/(dx^4) + sin(y^("')) = 0

Q 1 | Page 382 | view solution

Determine order and degree(if defined) of differential equation  y' + 5y = 0

Q 2 | Page 382 | view solution

Determine order and degree(if defined) of differential equation ((ds)/(dt))^4 + 3s  (d^2s)/(dt^2) = 0

Q 3 | Page 382 | view solution

Determine order and degree(if defined) of differential equation (d^2y)/(dx^2)^2 + cos(dy/dx) = 0

Q 4 | Page 382 | view solution

Determine order and degree(if defined) of differential equation (d^2y)/(dx^2) = cos 3x + sin 3x

Q 5 | Page 382 | view solution

Determine order and degree(if defined) of differential equation

( y′′′) + (y″)3 + (y′)4 + y5 = 0

Q 6 | Page 382 | view solution

Determine order and degree(if defined) of differential equation y′′′ + 2y″ + y′ = 0

Q 7 | Page 382 | view solution

Determine order and degree(if defined) of differential equation y′ + y = ex

Q 8 | Page 383 | view solution

Determine order and degree(if defined) of differential equation  y″ + (y′)2 + 2y = 0

Q 9 | Page 383 | view solution

Determine order and degree(if defined) of differential equation  y″ + 2y′ + sin y = 0

Q 10 | Page 383 | view solution

The degree of the differential equation

((d^2y)/(dx^2))^3 + ((dy)/(dx))^2 + sin ((dy)/(dx)) + 1 = 0

(A) 3

(B) 2

(C) 1

(D) not defined

Q 11 | Page 383 | view solution

The order of the differential equation

2x^2 (d^2y)/(dx^2) - 3 (dy)/(dx) + y = 0 is

(A) 2

(B) 1

(C) 0

(D) not defined

Q 12 | Page 383 | view solution

#### Page 385

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = ex + 1  :  y″ – y′ = 0

Q 1 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = x2 + 2x + C  :  y′ – 2x – 2 = 0

Q 2 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = cos x + C : y′ + sin x = 0

Q 3 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y sqrt(1 + x^2) : y' = (xy)/(1+x^2)

Q 4 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = Ax   :  xy′ = y (x ≠ 0)

Q 5 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = x sin x  : xy′ = y + xsqrt(x^2 - y^2) (x != 0 and x > y, or x < -y)

Q 6 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

xy = log y + C :  y' = (y^2)/(1 - xy) (xy != 1)

Q 7 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:

y – cos y = x :  (y sin y + cos y + x) y′ = y

Q 8 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

x + y = tan–1y   :   y2 y′ + y2 + 1 = 0

Q 9 | Page 385 | view solution

verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation

y = sqrt(a^2 - x^2 ) x in (-a,a)     :     x + y  dy/dx = 0(y != 0)

Q 10 | Page 385 | view solution

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

Q 11 | Page 385 | view solution

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

Q 12 | Page 385 | view solution

#### Page 391

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

x/a + y/b = 1

Q 1 | Page 391 | view solution

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y2 = a (b2 – x2)

Q 2 | Page 391 | view solution

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = a e3x + b e– 2x

Q 3 | Page 391 | view solution

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = e2x (a + bx)

Q 4 | Page 391 | view solution

Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.

y = ex (a cos x + b sin x)

Q 5 | Page 391 | view solution

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

Q 6 | Page 391 | view solution

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

Q 7 | Page 391 | view solution

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

Q 8 | Page 391 | view solution

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

Q 9 | Page 391 | view solution

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

Q 10 | Page 391 | view solution

Which of the following differential equations has y = c1 ex + c2 e–x as the general solution?

(A) (d^2y)/(dx^2) + y = 0

(B) (d^2y)/(dx^2) - y = 0

(C) (d^2y)/(dx^2) + 1 = 0

(D) (d^2y)/(dx^2)  - 1 = 0

Q 11 | Page 391 | view solution

Which of the following differential equation has y = x as one of its particular solution?

A. (d^2y)/(dx^2) - x^2 (dy)/(dx) + xy = x

B. (d^2y)/(dx^2) + x dy/dx + xy = x

C. (d^2y)/(dx^2) - x^2 dy/dx + xy = 0

D. (d^2y)/(dx^2) + x dy/dx + xy = 0

Q 12 | Page 391 | view solution

#### Pages 395 - 397

For the differential equations find the general solution:

dy/dx = (1 - cos x)/(1+cos x)

Q 1 | Page 395 | view solution

For the differential equations find the general solution:

dy/dx = sqrt(4-y^2)      (-2 < y < 2)

Q 2 | Page 395 | view solution

For the differential equations find the general solution:

dy/dx + y = 1(y != 1)

Q 3 | Page 396 | view solution

For the differential equations find the general solution:

sec2 x tan y dx + sec2 y tan x dy = 0

Q 4 | Page 396 | view solution

For the differential equations find the general solution:

(ex + e–x) dy – (ex – e–x) dx = 0

Q 5 | Page 396 | view solution

For the differential equations find the general solution:

dy/dx = (1+x^2)(1+y^2)

Q 6 | Page 396 | view solution

For the differential equations find the general solution:

y log y dx – x dy = 0

Q 7 | Page 396 | view solution

For the differential equations find the general solution:

x^5 dy/dx = - y^5

Q 8 | Page 396 | view solution

For the differential equations find the general solution:

dy/dx = sin^(-1) x

Q 9 | Page 396 | view solution

For the differential equations find the general solution:

ex tan y dx + (1 – ex) sec2 y dy = 0

Q 10 | Page 396 | view solution

For each of the differential equations find a particular solution satisfying the given condition:

(x^3 + x^2 + x + 1) dy/dx = 2x^2 + x; y = 1 when x = 0

Q 11 | Page 396 | view solution

For each of the differential equations find a particular solution satisfying the given condition:

x(x^2 - 1) dy/dx = 1 , y = 0 " when x " = 2

Q 12 | Page 396 | view solution

For each of the differential equations find a particular solution satisfying the given condition:

cos (dx/dy) = a(a in R); y = 1 when x = 0

Q 13 | Page 396 | view solution

For each of the differential equations find a particular solution satisfying the given condition:

dy/dx = y tan x; y = 1 when x = 0

Q 14 | Page 396 | view solution

Find the equation of a curve passing through the point (0, 0) and whose differential equation is y′ = e x sin x.

Q 15 | Page 396 | view solution

For the differential equation xy(dy)/(dx) = (x + 2)(y + 2)  find the solution curve passing through the point (1, –1).

Q 16 | Page 396 | view solution

Find the equation of a curve passing through the point (0, –2) given that at any point (x ,y) 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.

Q 17 | Page 396 | view solution

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).

Q 18 | Page 396 | view solution

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.

Q 19 | Page 396 | view solution

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).

Q 20 | Page 397 | view solution

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 (e0.5 = 1.648)..

Q 21 | Page 397 | view solution

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?

Q 22 | Page 397 | view solution

The general solution of the differential equation dy/dx = e^(x+y) is

(A) ex + e–y = C

(B) ex + ey = C

(C) e–x + ey = C

(D) e–x + e–y = C

Q 23 | Page 397 | view solution

#### Pages 406 - 407

Show that the given differential equation is homogeneous and solve each of them

(x2 + xy) dy = (x2 + y2) dx

Q 1 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve each of them

y' = (x + y)/x

Q 2 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve each of them

(x – y) dy – (x + y) dx = 0

Q 3 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve each of them

(x2 – y2) dx + 2xy dy = 0

Q 4 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve each of them

x^2 dy/dx = x^2 - 2y^2 + xy

Q 5 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve each of them

xdy - ydx =  sqrt(x^2 + y^2) dx

Q 6 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve each of them

{xcos(y/x) + ysin(y/x)}ydx = {ysin (y/x) -  xcos(y/x)}xdy

Q 7 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve

x dy/dx - y +  x sin (y/x) = 0

Q 8 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve

ydx + xlog(y/x)dy - 2xdy = 0

Q 9 | Page 406 | view solution

Show that the given differential equation is homogeneous and solve

(1+e^(x/y))dx + e^(x/y) (1 - x/y)dy = 0

Q 10 | Page 406 | view solution

For the differential equations find the particular solution satisfying the given condition:

(x + y) dy + (x – y) dx = 0; y = 1 when x = 1

Q 11 | Page 406 | view solution

For the differential equations find the particular solution satisfying the given condition:

x2 dy + (xy + y2) dx = 0; y = 1 when x = 1

Q 12 | Page 406 | view solution

For the differential equations find the particular solution satisfying the given condition:

[xsin^2(y/x - y)] dx + xdy = 0; y = pi/4 when x = 1

Q 13 | Page 406 | view solution

For the differential equations find the particular solution satisfying the given condition:

dy/dx -  y/x + cosec (y/x) = 0; y = 0 when x = 1

Q 14 | Page 406 | view solution

For the differential equations find the particular solution satisfying the given condition:

2xy + y^2 - 2x^2  dy/dx = 0; y = 2  " when x " = 1

Q 15 | Page 406 | view solution

A homogeneous differential equation of the from dx/dy = h(x/y) can be solved by making the substitution

A. y = vx

B. v = yx

C. vy

D. x = v

Q 16 | Page 406 | view solution

Which of the following is a homogeneous differential equation?

(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0

(B) (xy) dx – (x3 + y3) dy = 0

(C) (x3 + 2y2) dx + 2xy dy = 0

(D) y2 dx + (x2 – xy – y2) dy = 0

Q 17 | Page 407 | view solution

#### Pages 413 - 414

For the differential equations find the general solution:

dy/dx  + 2y = sin x

Q 1 | Page 413 | view solution

For the differential equations find the general solution:

dy/dx + 3y = e^(-2x)

Q 2 | Page 413 | view solution

For the differential equations find the general solution:

dy/dx + y/x = x^2

Q 3 | Page 413 | view solution

For the differential equations find the general solution:

dy/dx + secxy = tan x (0 <= x < pi/2)

Q 4 | Page 413 | view solution

For the differential equations find the general solution:

cos^2 x dy/dx + y = tan x(0 <= x < pi/2)

Q 5 | Page 413 | view solution

For the differential equations find the general solution:

x dy/dx +  2y= x^2 log x

Q 6 | Page 413 | view solution

For the differential equations find the general solution:

x log x dy/dx + y=    2/x log x

Q 7 | Page 413 | view solution

For the differential equations find the general solution: (1 + x2) dy + 2xy dx = cot x dx (x ≠ 0)

Q 8 | Page 413 | view solution

For the differential equations find the general solution:

x dy/dx + y - x + xycot x = 0(x != 0)

Q 9 | Page 414 | view solution

For the differential equations find the general solution:

(x + y) dy/dx = 1

Q 10 | Page 414 | view solution

For the differential equations find the general solution:

y dx + (x – y2) dy = 0

Q 11 | Page 414 | view solution

For the differential equations find the general solution:

(x + 3y^2) dy/dx = y(y > 0)

Q 12 | Page 414 | view solution

For the differential equations given find a particular solution satisfying the given condition:

dy/dx + 2y tan x = sin x; y = 0 " when x " = pi/3

Q 13 | Page 414 | view solution

For the differential equations given find a particular solution satisfying the given condition:

(1 + x^2)dy/dx + 2xy = 1/(1 + x^2); y = 0 " when x " =1

Q 14 | Page 414 | view solution

For the differential equations given find a particular solution satisfying the given condition:

dy/dx - 3ycotx = sin 2x; y = 2 " when x  "= pi/2

Q 15 | Page 414 | view solution

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

Q 16 | Page 414 | view solution

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.

Q 17 | Page 414 | view solution

The Integrating Factor of the differential equation dy/dx - y = 2x^2 is

A. ex

B. ey

C. 1/x

D. x

Q 18 | Page 414 | view solution

The integrating factor of the differential equation.

(1 - y^2) dx/dy + yx = ay(-1 < < 1) is

Q 19 | Page 414 | view solution

#### Pages 419 - 421

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

(d^2y)/dx^2 + 5x(dy/dx)^2 - 6y = log x

Q 1.1 | Page 419 | view solution

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

((dy)/(dx))^3 -4(dy/dx)^2 + 7y = sin x

Q 1.2 | Page 419 | view solution

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

(d^4y)/dx^4 - sin ((d^3y)/(dx^3)) = 0

Q 1.3 | Page 419 | view solution

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

y = a ex + b e–x + x2    :  x (d^2y)/(dx62) + 2 dy/dx - xy + x^2 - 2 = 0

Q 2.1 | Page 420 | view solution

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

y = e^x (acos x + b sin x)  :  (d^2y)/(dx^2) - 2 dy/dx + 2y = 0

Q 2.2 | Page 420 | view solution

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

y = xsin 3x   :   (d^2y)/(dx^2) + 9y - 6 cos 3x = 0

Q 2.3 | Page 420 | view solution

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

x^2 = 2y^2 log y    :  (x^2  + y^2) dy/dx - xy = 0

Q 2.4 | Page 420 | view solution

Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.

Q 3 | Page 420 | view solution

Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation  (x3 – 3x y2) dx = (y3 – 3x2y) dy, where c is a parameter.

Q 4 | Page 420 | view solution

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

Q 4 | Page 420 | view solution

Find the general solution of the differential equation  dy/dx + sqrt((1-y^2)/(1-x^2)) = 0

Q 6 | Page 420 | view solution

Show that the general solution of the differential equation  dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0 is given by (x + + 1) = (1 – – y – 2xy), where is parameter

Q 7 | Page 420 | view solution

Find the equation of the curve passing through the point (0,pi/4), whose differential equation is sin x cos y dx + cos x sin y dy = 0.

Q 8 | Page 420 | view solution

Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.

Q 9 | Page 420 | view solution

Solve the differential equation ye^(x/y) dx = (xe^(x/y) + y^2)dy (y != 0)

Q 10 | Page 420 | view solution

Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)

Q 11 | Page 420 | view solution

Solve the differential equation [e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0)

Q 12 | Page 421 | view solution

Find a particular solution of the differential equation dy/dx + y cot x = 4xcosec x(x != 0), given that y = 0 when x = pi/2

Q 13 | Page 421 | view solution

Find a particular solution of the differential equation(x + 1) dy/dx = 2e^(-y) - 1, given that y = 0 when x = 0

Q 14 | Page 421 | view solution

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?

Q 15 | Page 421 | view solution

The general solution of the differential equation (ydx - xdy)/y = 0

A. xy = C

B. = Cy2

C. = Cx

D. y = Cx2

Q 16 | Page 421 | view solution

The general solution of a differential equation of the type  dx/dy + P_1 x = Q is

(A) y e^(intP_1 dy) = int(Q_1 e^(intP_1 dy)) dy + C

(B) y . e^(intP_1 dx) = int(Q_1 e^(intP_1 dx)) dx + C

(C) x e^(intP_1 dy) = int(Q_1 e^(intP_1 dy)) dy + C

(D) xe^(intP_1 dx) = int(Q_1 e^(intP_1 dx)) dx + C`

Q 17 | Page 421 | view solution

The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is

A. xey + x2 = C

B. xey + y2 = C

C. yex + x2 = C

D. yey x2 = C

Q 18 | Page 421 | view solution