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NCERT solutions Mathematics Class 12 Part 2 chapter 9 Differential Equations

Chapters

NCERT Solutions for Mathematics Class 12 Part 1

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2

Chapter 9 - 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

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

Q 2 | Page 382

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

Q 3 | Page 382

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

Q 4 | Page 382

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

Q 5 | Page 382

Determine order and degree(if defined) of differential equation  

( y′′′) + (y″)3 + (y′)4 + y5 = 0

Q 6 | Page 382

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

Q 7 | Page 382

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

Q 8 | Page 383

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

Q 9 | Page 383

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

Q 10 | Page 383

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Q 6 | Page 391

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

Q 7 | Page 391

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

Q 8 | Page 391

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

Q 9 | Page 391

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

 
Q 10 | Page 391

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

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

Pages 395 - 397

For the differential equations find the general solution:

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

Q 1 | Page 395

For the differential equations find the general solution:

`dy/dx = sqrt(4-y^2)      (-2 < y < 2)`

Q 2 | Page 395

For the differential equations find the general solution:

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

Q 3 | Page 396

For the differential equations find the general solution:

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

Q 4 | Page 396

For the differential equations find the general solution:

(ex + e–x) dy – (ex – e–x) dx = 0

Q 5 | Page 396

For the differential equations find the general solution:

`dy/dx = (1+x^2)(1+y^2)`

Q 6 | Page 396

For the differential equations find the general solution:

y log y dx – x dy = 0

Q 7 | Page 396

For the differential equations find the general solution:

`x^5 dy/dx = - y^5`

Q 8 | Page 396

For the differential equations find the general solution:

`dy/dx = sin^(-1) x`

Q 9 | Page 396

For the differential equations find the general solution:

ex tan y dx + (1 – ex) sec2 y dy = 0

Q 10 | Page 396

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

`y' = (x + y)/x`

Q 2 | Page 406

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

(x – y) dy – (x + y) dx = 0

Q 3 | Page 406

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

(x2 – y2) dx + 2xy dy = 0

Q 4 | Page 406

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

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

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

Show that the given differential equation is homogeneous and solve 

`x dy/dx - y +  x sin (y/x) = 0`

Q 8 | Page 406

Show that the given differential equation is homogeneous and solve 

`ydx + xlog(y/x)dy - 2xdy = 0`

 

Q 9 | Page 406

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

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

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

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

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

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

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

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

Pages 413 - 414

For the differential equations find the general solution:

`dy/dx  + 2y = sin x`

Q 1 | Page 413

For the differential equations find the general solution:

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

Q 2 | Page 413

For the differential equations find the general solution:

`dy/dx + y/x = x^2`

Q 3 | Page 413

For the differential equations find the general solution:

`dy/dx + secxy = tan x (0 <= x < pi/2)`

Q 4 | Page 413

For the differential equations find the general solution:

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

Q 5 | Page 413

For the differential equations find the general solution:

`x dy/dx +  2y= x^2 log x`

Q 6 | Page 413

For the differential equations find the general solution:

`x log x dy/dx + y=    2/x log x`

Q 7 | Page 413

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

Q 8 | Page 413

For the differential equations find the general solution:

`x dy/dx + y - x + xycot x = 0(x != 0)`

Q 9 | Page 414

For the differential equations find the general solution:

`(x + y) dy/dx = 1`

Q 10 | Page 414

For the differential equations find the general solution:

y dx + (x – y2) dy = 0

 

Q 11 | Page 414

For the differential equations find the general solution:

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

Q 12 | Page 414

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

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

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

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

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

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

The integrating factor of the differential equation.

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

Q 19 | Page 414

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

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

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

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

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

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

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

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

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

Q 4 | Page 420

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

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

Q 6 | Page 420

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

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

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

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

Q 10 | Page 420

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

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

Q 12 | Page 421

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

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

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

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

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

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

NCERT Mathematics Class 12 Part 2

Mathematics Textbook for Class 12 Part 2
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