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NCERT solutions for Mathematics Exemplar Class 12 chapter 9 - Differential Equations [Latest edition]

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Mathematics Exemplar Class 12 - Shaalaa.com
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Chapter 9: Differential Equations

Solved ExamplesExercise
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Solved Examples [Pages 180 - 192]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 9 Differential EquationsSolved Examples [Pages 180 - 192]

Short Answer

Solved Examples | Q 1 | Page 180

Find the differential equation of the family of curves y = Ae2x + B.e–2x.

Solved Examples | Q 2 | Page 181

Find the general solution of the differential equation `"dy"/"dx" = y/x`.

Solved Examples | Q 3 | Page 181

Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.

Solved Examples | Q 4 | Page 181

Solve the differential equation `"dy"/"dx" + y/x` = x2.

Solved Examples | Q 5 | Page 182

Find the differential equation of the family of lines through the origin.

Solved Examples | Q 6 | Page 182

Find the differential equation of all non-horizontal lines in a plane.

Solved Examples | Q 7 | Page 182

Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.

Long Answer

Solved Examples | Q 8 | Page 183

Find the equation of a curve passing through the point (1, 1) if the perpendicular distance of the origin from the normal at any point P(x, y) of the curve is equal to the distance of P from the x-axis.

Solved Examples | Q 9 | Page 184

Find the equation of a curve passing through `(1, pi/4)` if the slope of the tangent to the curve at any point P(x, y) is `y/x - cos^2  y/x`.

Solved Examples | Q 10 | Page 185

Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`

Solved Examples | Q 11 | Page 186

State the type of the differential equation for the equation. xdy – ydx = `sqrt(x^2 + y^2)  "d"x` and solve it

Objective Type Questions from 12 to 21

Solved Examples | Q 12 | Page 187

The degree of the differential equation `(1 + "dy"/"dx")^3 = (("d"^2y)/("d"x^2))^2` is ______.

  • 1

  • 2

  • 3

  • 4

Solved Examples | Q 13 | Page 187

The degree of the differential equation `("d"^2y)/("d"x^2) + 3("dy"/"dx")^2 = x^2 log(("d"^2y)/("d"x^2))` is ______.

  • 1

  • 2

  • 3

  • not defined

Solved Examples | Q 14 | Page 187

The order and degree of the differential equation `[1 + ("dy"/"dx")^2]^2 = ("d"^2y)/("d"x^2)` respectively, are ______.

  • 1, 2

  • 2, 2

  • 2, 1

  • 4, 2

Solved Examples | Q 15 | Page 187

The order of the differential equation of all circles of given radius a is ______.

  • 1

  • 2

  • 3

  • 4

Solved Examples | Q 16 | Page 187

The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.

  • Straight lines

  • Circles

  • Parabolas

  • Ellipses

Solved Examples | Q 17 | Page 188

The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.

  • ex 

  • log x

  • log (log x)

  • x

Solved Examples | Q 18 | Page 188

A solution of the differential equation `("dy"/"dx")^2 - x "dy"/"dx" + y` = 0 is ______.

  • y = 2

  • y = 2x

  • y = 2x – 4

  • y = 2x2 – 4

Solved Examples | Q 19 | Page 188

Which of the following is not a homogeneous function of x and y.

  • x2 + 2xy

  • 2x – y

  • `cos^2 (y/x) + y/x`

  • sinx – cosy

Solved Examples | Q 20 | Page 188

Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.

  • `1/x + 1/y` = c

  • logx . logy = c

  • xy = c

  • x + y = c

Solved Examples | Q 21 | Page 189

The solution of the differential equation `x "dt"/"dx" + 2y` = x2 is ______.

  • y = `(x^2 + "c")/(4x^2)`

  • y = `x^2/4 + "c"`

  • y = `(x^4 + "c")/x^2`

  • y = `(x^4 + "c")/(4x^2)`

Fill in the blanks of the following:

Solved Examples | Q 22. (i) | Page 188

Order of the differential equation representing the family of parabolas y2 = 4ax is ______.

Solved Examples | Q 22. (ii) | Page 188

The degree of the differential equation `("dy"/"dx")^2 + (("d"^2y)/("d"x^2))^2` = 0 is ______.

Solved Examples | Q 22. (iii) | Page 189

The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.

Solved Examples | Q 22. (iv) | Page 189

F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.

Solved Examples | Q 22. (v) | Page 189

An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.

Solved Examples | Q 22. (vi) | Page 189

Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.

Solved Examples | Q 22. (vii) | Page 189

The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.

Solved Examples | Q 22. (viii) | Page 190

The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.

Solved Examples | Q 22. (ix) | Page 190

The differential equation representing the family of curves y = A sinx + B cosx is ______.

Solved Examples | Q 22. (x) | Page 190

`("e"^(-2sqrt(x))/sqrt(x) - y/sqrt(x))("d"x)/("d"y) = 1(x ≠ 0)` when written in the form `"dy"/"dx" + "P"y` = Q, then P = ______.

State whether the following statement is True or False

Solved Examples | Q 23. (i) | Page 191

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

  • True

  • False

Solved Examples | Q 23. (ii) | Page 191

Degree of the differential equation `sqrt(1 + ("d"^2y)/("d"x^2)) = x + "dy"/"dx"` is not defined.

  • True

  • False

Solved Examples | Q 23. (iii) | Page 191

`"dy"/"dx" + y` = 5 is a differential equation of the type `"dy"/"dx" + "P"y` = Q but it can be solved using variable separable method also.

  • True

  • False

Solved Examples | Q 23. (iv) | Page 191

F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.

  • True

  • False

Solved Examples | Q 23. (v) | Page 191

F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.

  • True

  • False

Solved Examples | Q 23. (vi) | Page 191

Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.

  • True

  • False

Solved Examples | Q 23. (vii) | Page 191

The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.

  • True

  • False

Solved Examples | Q 23. (viii) | Page 191

The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.

  • True

  • False

Solved Examples | Q 23. (ix) | Page 191

x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.

  • True

  • False

Solved Examples | Q 23. (x) | Page 192

y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.

  • True

  • False

Exercise [Pages 193 - 203]

NCERT solutions for Mathematics Exemplar Class 12 Chapter 9 Differential EquationsExercise [Pages 193 - 203]

Short Answer

Exercise | Q 1 | Page 193

Find the solution of `"dy"/"dx"` = 2y–x.

Exercise | Q 2 | Page 193

Find the differential equation of all non-vertical lines in a plane.

Exercise | Q 3 | Page 193

Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.

Exercise | Q 4 | Page 193

Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.

Exercise | Q 5 | Page 193

Solve the differential equation `"dy"/"dx" + 2xy` = y

Exercise | Q 6 | Page 193

Find the general solution of `"dy"/"dx" + "a"y` = emx 

Exercise | Q 7 | Page 193

Solve the differential equation `"dy"/"dx" + 1` = ex + y.

Exercise | Q 8 | Page 193

Solve: ydx – xdy = x2ydx.

Exercise | Q 9 | Page 193

Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.

Exercise | Q 10 | Page 193

Find the general solution of `(x + 2y^3)  "dy"/"dx"` = y

Exercise | Q 11 | Page 193

If y(x) is a solution of `((2 + sinx)/(1 + y))"dy"/"dx"` = – cosx and y (0) = 1, then find the value of `y(pi/2)`.

Exercise | Q 12 | Page 193

If y(t) is a solution of `(1 + "t")"dy"/"dt" - "t"y` = 1 and y(0) = – 1, then show that y(1) = `-1/2`.

Exercise | Q 13 | Page 194

Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.

Exercise | Q 14 | Page 194

Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.

Exercise | Q 15 | Page 194

Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2 

Exercise | Q 16 | Page 194

Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.

Exercise | Q 17 | Page 194

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

Exercise | Q 18 | Page 194

Find the general solution of y2dx + (x2 – xy + y2) dy = 0.

Exercise | Q 19 | Page 194

Solve: (x + y)(dx – dy) = dx + dy. [Hint: Substitute x + y = z after seperating dx and dy]

Exercise | Q 20 | Page 194

Solve: `2(y + 3) - xy "dy"/"dx"` = 0, given that y(1) = – 2.

Exercise | Q 21 | Page 194

Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`

Exercise | Q 22 | Page 194

Form the differential equation by eliminating A and B in Ax2 + By2 = 1

Exercise | Q 23 | Page 194

Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.

Exercise | Q 24 | Page 194

Find the differential equation of system of concentric circles with centre (1, 2).

Long Answer

Exercise | Q 25 | Page 194

Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`

Exercise | Q 26 | Page 194

Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.

Exercise | Q 27 | Page 194

Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]

Exercise | Q 28 | Page 194

Find the general solution of `("d"y)/("d"x) -3y = sin2x`

Exercise | Q 29 | Page 194

Find the equation of a curve passing through (2, 1) if the slope of the tangent to the curve at any point (x, y) is `(x^2 + y^2)/(2xy)`.

Exercise | Q 30 | Page 195

Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x, y) is `(y - 1)/(x^2 + x)`

Exercise | Q 31 | Page 195

Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point (x, y) is equal to the square of the difference of the abcissa and ordinate of the point.

Exercise | Q 32 | Page 195

Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P(x, y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB.

Exercise | Q 33 | Page 195

Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`

Objective Type from 34 to 75

Exercise | Q 34 | Page 195

The degree of the differential equation `(("d"^2y)/("d"x^2))^2 + (("d"y)/("d"x))^2 = xsin(("d"y)/("d"x))` is ______.

  • 1

  • 2

  • 3

  • Not defined

Exercise | Q 35 | Page 195

The degree of the differential equation `[1 + (("d"y)/("d"x))^2]^(3/2) = ("d"^2y)/("d"x^2)` is ______.

  • 4

  • `3/2`

  • not defined

  • 2

Exercise | Q 36 | Page 195

The order and degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^(1/4) + x^(1/5)` = 0, respectively, are ______.

  • 2 and not defined

  • 2 and 2

  • 2 and 3

  • 3 and 3

Exercise | Q 37 | Page 195

If y = e–x (Acosx + Bsinx), then y is a solution of ______.

  • `("d"^2y)/("d"x^2) + 2("d"y)/("d"x)` = 0

  • `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y ` = 0

  • `("d"^2y)/("d"x^2) + 2 ("d"y)/("d"x) + 2y` = 0

  • `("d"^2y)/("d"x^2) + 2y` = 0

Exercise | Q 38 | Page 196

The differential equation for y = Acos αx + Bsin αx, where A and B are arbitrary constants is ______.

  • `("d"^2y)/("d"x^2) - alpha^2y` = 0

  • `("d"^2y)/("d"x^2) + alpha^2y` = 0

  • `("d"^2y)/("d"x^2) + alphay` = 0

  • `("d"^2y)/("d"x^2) - alphay` = 0

Exercise | Q 39 | Page 196

Solution of differential equation xdy – ydx = 0 represents : ______.

  • A rectangular hyperbola

  • Parabola whose vertex is at origin

  • Straight line passing through origin

  • A circle whose centre is at origin

Exercise | Q 40 | Page 196

Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.

  • cosx

  • tanx

  • secx

  • sinx

Exercise | Q 41 | Page 196

Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.

  • tanx + tany = k

  • tanx – tany = k

  • `tanx/tany` = k

  • tanx . tany = k

Exercise | Q 42 | Page 196

Family y = Ax + A3 of curves is represented by the differential equation of degree ______.

  • 1

  • 2

  • 3

  • 4

Exercise | Q 43 | Page 196

Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.

  • x

  • logx

  • `1/x`

  • – x

Exercise | Q 44 | Page 196

Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.

  • xy = – ex

  • xy = – e-x 

  • xy = – 1

  • y = 2ex – 1

Exercise | Q 45 | Page 197

The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______. 

  • None

  • One

  • Two

  • Infinite

Exercise | Q 46 | Page 197

Which of the following is a second order differential equation?

  • (y′)2 + x = y2

  • y′y′′+ y = sin x

  • y″ + (y'')2 + y = 0

  • y′ = y2 

Exercise | Q 47 | Page 197

Integrating factor of the differential equation `(1 - x^2) ("d"y)/("d"x) - xy` = 1 is ______.

  • – x

  • `x/(1 + x^2)`

  • `sqrt(1 - x^2)`

  • `1/2 log (1 - x^2)`

Exercise | Q 48 | Page 197

tan–1x + tan–1y = c is the general solution of the differential equation ______.

  • `("d"y)/("d"x) = (1 + y^2)/(1 + x^2)`

  • `("d"y)/("d"x) = (1 + x^2)/(1 + y^2)`

  • (1 + x2)dy + (1 + y2)dx = 0

  • (1 + x2)dx + (1 + y2)dy = 0

Exercise | Q 49 | Page 197

The differential equation `y ("d"y)/("d"x) + "c"` represents: ______.

  • Family of hyperbolas

  • Family of parabolas

  • Family of ellipses

  • Family of circles

Exercise | Q 50 | Page 197

The general solution of ex cosy dx – ex siny dy = 0 is ______.

  • ex cosy = k

  • ex siny = k

  • ex = k cosy

  • ex = k siny

Exercise | Q 51 | Page 197

The degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^3 + 6y^5` = 0 is ______.

  • 1

  • 2

  • 3

  • 5

Exercise | Q 52 | Page 197

The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.

  • y = ex(x – 1)

  • y = xe–x 

  • y = xe–x + 1

  • y = (x + 1)e–x 

Exercise | Q 53 | Page 198

Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.

  • cosx

  • secx

  • ecosx

  • esecx

Exercise | Q 54 | Page 198

The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.

  • y = tan–1x

  • y – x = k(1 + xy)

  • x = tan–1y

  • tan(xy) = k

Exercise | Q 55 | Page 198

The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.

  • `x/"e"^x`

  • `"e"^x/x`

  • xex 

  • ex 

Exercise | Q 56 | Page 198

y = aemx+ be–mx satisfies which of the following differential equation?

  • `("d"y)/("d"x) + "m"y` = 0

  • `("d"y)/("d"x) - "m"y` = 0

  • `("d"^2y)/("d"x^2) - "m"^2y` = 0

  • `("d"^2y)/("d"x^2) + "m"^2y` = 0

Exercise | Q 57 | Page 198

The solution of the differential equation cosx siny dx + sinx cosy dy = 0 is ______.

  • `sinx/siny` = c

  • sinx siny = c

  • sinx + siny = c

  • cosx cosy = c

Exercise | Q 58 | Page 198

The solution of `x ("d"y)/("d"x) + y` = ex is ______.

  • y = `"e"^x/x + "k"/x`

  • y = xex + cx

  • y = xex + k

  • x = `"e"^y/y + "k"/y`

Exercise | Q 59 | Page 199

The differential equation of the family of curves x2 + y2 – 2ay = 0, where a is arbitrary constant, is ______.

  • `(x^2 - y^2) ("d"y)/("d"x)` = 2xy

  • `2(x^2 + y^2) ("d"y)/("d"x)` = xy

  • `2(x^2 - y^2) ("d"y)/("d"x)` = xy

  • `(x^2 + y^2) ("d"y)/("d"x)` = 2xy

Exercise | Q 60 | Page 199

Family y = Ax + A3 of curves will correspond to a differential equation of order ______.

  • 3

  • 2

  • 1

  • Not defined

Exercise | Q 61 | Page 199

The general solution of `("d"y)/("d"x) = 2x"e"^(x^2 - y)` is ______.

  • `"e"^(x^2 - y)` = c

  • `"e"^-y + "e"^(x^2)` = c

  • `"e"^-y = "e"^(x^2)` + c

  • `"e"^(x^2 + y)` = c

Exercise | Q 62 | Page 199

The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is ______.

  • An ellipse

  • Parabola

  • Circle

  • Rectangular hyperbola

Exercise | Q 63 | Page 199

The general solution of the differential equation `("d"y)/("d"x) = "e"^(x^2/2) + xy` is ______.

  • y = `"ce"^((-x^2)/2`

  • y = `"ce"^((x^2)/2`

  • y = `(x + "c")"e"^((x^2)/2`

  • y = `("c" - x)"e"^((x^2)/2`

Exercise | Q 64 | Page 199

The solution of the equation (2y – 1)dx – (2x + 3)dy = 0 is ______.

  • `(2x - 1)/(2y + 3)` = k

  • `(y + 1)/(2x - 3)` = k

  • `(2x + 3)/(2y - 1)` = k

  • `(2x - 1)/(2y - 1)` = k

Exercise | Q 65 | Page 200

The differential equation for which y = acosx + bsinx is a solution, is ______.

  • `("d"^2y)/("d"x^2) + y` = 0

  • `("d"^2y)/("d"x^2) - y` = 0

  • `("d"^2y)/("d"x^2) + ("a" + "b")y` = 0

  • `("d"^2y)/("d"x^2) + ("a" - "b")y` = 0

Exercise | Q 66 | Page 200

The solution of `("d"y)/("d"x) + y = "e"^-x`, y(0) = 0 is ______.

  • y = `"e"^x (x - 1)`

  • y = xex

  • y = `x"e"^-x + 1`

  • y = xe–x 

Exercise | Q 67 | Page 200

The order and degree of the differential equation `(("d"^3y)/("d"x^3))^2 - 3 ("d"^2y)/("d"x^2) + 2(("d"y)/("d"x))^4` = y4 are ______.

  • 1, 4

  • 3, 4

  • 3, 4

  • 3, 2

Exercise | Q 68 | Page 200

The order and degree of the differential equation `[1 + (("d"y)/("d"x))^2] = ("d"^2y)/("d"x^2)` are ______.

  • `2, 3/2`

  • 2, 3

  • 2, 1

  • 3, 4

Exercise | Q 69 | Page 200

The differential equation of the family of curves y2 = 4a(x + a) is ______.

  • `y^2 - 4 ("d"y)/("d"x)(x + ("d"y)/("d"x))`

  • `2y ("d"y)/("d"x)` = 4a

  • `y ("d"^2y)/("d"x^2) + (("d"y)/("d"x))^2` = 0

  • `2x ("d"y)/("d"x) + y(("d"y)/("d"x))^2 - y`

Exercise | Q 70 | Page 200

Which of the following is the general solution of `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0?

  • y = (Ax + B)ex

  • y = (Ax + B)e–x

  • y = Aex + Be–x

  • y = Acosx + Bsinx

Exercise | Q 71 | Page 201

General solution of `("d"y)/("d"x) + ytanx = secx` is ______.

  • y secx = tanx + c

  • y tanx = secx + c

  • tanx = y tanx + c

  • x secx = tany + c

Exercise | Q 72 | Page 201

Solution of the differential equation `("d"y)/("d"x) + y/x` = sec x is ______.

  • x(y + cosx) = sinx + c

  • x(y – cosx) = sinx + c

  • xy cosx = sinx + c

  • x(y + cosx) = cosx + c

Exercise | Q 73 | Page 201

The general solution of the differential equation (ex + 1) ydy = (y + 1) exdx is ______.

  • (y + 1) = k(ex + 1)

  • y + 1 = ex + 1 + k

  • y = log {k(y + 1)(ex + 1)}

  • y = `log{("e"^x + 1)/(y + 1)} + "k"`

Exercise | Q 74 | Page 201

The solution of the differential equation `("d"y)/("d"x) = "e"^(x - y) + x^2 "e"^-y` is ______.

  • y =`"e"^(x - y) = x^2 "e"^-y + "c"`

  • `"e"^y - "e"^x = x^3/3 + "c"`

  • `"e"^x + "e"^y = x^3/3 + "c"`

  • `"e"^x - "e"^y = x^3/3 + "c"`

Exercise | Q 75 | Page 201

The solution of the differential equation `("d"y)/("d"x) + (2xy)/(1 + x^2) = 1/(1 + x^2)^2` is ______.

  • y(1 + x2) = c + tan–1x

  • `y/(1 + x^2) = "c" + tan^-1x`

  • y log(1 + x2) = c + tan–1x

  • y(1 + x2) = c + sin–1x

Fill in the blanks of the following (i to xi)

Exercise | Q 76.(i) | Page 201

The degree of the differential equation `("d"^2y)/("d"x^2) + "e"^((dy)/(dx))` = 0 is ______.

Exercise | Q 76.(ii) | Page 201

The degree of the differential equation `sqrt(1 + (("d"y)/("d"x))^2)` = x is ______.

Exercise | Q 76.(iii) | Page 202

The number of arbitrary constants in the general solution of a differential equation of order three is ______.

Exercise | Q 76.(iv) | Page 202

`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.

Exercise | Q 76.(v) | Page 202

General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.

Exercise | Q 76.(vi) | Page 202

The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.

Exercise | Q 76.(vii) | Page 202

The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.

Exercise | Q 76.(viii) | Page 202

The solution of the differential equation ydx + (x + xy)dy = 0 is ______.

Exercise | Q 76.(ix) | Page 202

General solution of `("d"y)/("d"x) + y` = sinx is ______.

Exercise | Q 76.(x) | Page 202

The solution of differential equation coty dx = xdy is ______.

Exercise | Q 76.(xi) | Page 202

The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.

State True or False for the following:

Exercise | Q 77.(i) | Page 202

Integrating factor of the differential equation of the form `("d"x)/("d"y) + "P"_1x = "Q"_1` is given by `"e"^(int P_1dy)`.

  • True

  • False

Exercise | Q 77.(ii) | Page 202

Solution of the differential equation of the type `("d"x)/("d"y) + "p"_1x = "Q"_1` is given by x.I.F. = `("I"."F") xx "Q"_1"d"y`.

  • True

  • False

Exercise | Q 77.(iii) | Page 203

Correct substitution for the solution of the differential equation of the type `("d"y)/("d"x) = "f"(x, y)`, where f(x, y) is a homogeneous function of zero degree is y = vx.

  • True

  • False

Exercise | Q 77.(iv) | Page 203

Correct substitution for the solution of the differential equation of the type `("d"x)/("d"y) = "g"(x, y)` where g(x, y) is a homogeneous function of the degree zero is x = vy.

  • True

  • False

Exercise | Q 77.(v) | Page 203

Number of arbitrary constants in the particular solution of a differential equation of order two is two.

  • True

  • False

Exercise | Q 77.(vi) | Page 203

The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.

  • True

  • False

Exercise | Q 77.(vii) | Page 203

The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.

  • True

  • False

Exercise | Q 77.(viii) | Page 203

Differential equation representing the family of curves y = ex (Acosx + Bsinx) is `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + 2y` = 0

  • True

  • False

Exercise | Q 77.(ix) | Page 203

The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.

  • True

  • False

Exercise | Q 77.(x) | Page 203

Solution of `x("d"y)/("d"x) = y + x tan  y/x` is `sin(y/x)` = cx

  • True

  • False

Exercise | Q 77.(xi) | Page 203

The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0

  • True

  • False

Chapter 9: Differential Equations

Solved ExamplesExercise
Mathematics Exemplar Class 12 - Shaalaa.com

NCERT solutions for Mathematics Exemplar Class 12 chapter 9 - Differential Equations

NCERT solutions for Mathematics Exemplar Class 12 chapter 9 (Differential Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 12 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 12 chapter 9 Differential Equations are Procedure to Form a Differential Equation that Will Represent a Given Family of Curves, Linear Differential Equations, Solutions of Linear Differential Equation, Homogeneous Differential Equations, Differential Equations with Variables Separable Method, Formation of a Differential Equation Whose General Solution is Given, General and Particular Solutions of a Differential Equation, Order and Degree of a Differential Equation, Basic Concepts of Differential Equation.

Using NCERT Class 12 solutions Differential Equations exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 12 prefer NCERT Textbook Solutions to score more in exam.

Get the free view of chapter 9 Differential Equations Class 12 extra questions for Mathematics Exemplar Class 12 and can use Shaalaa.com to keep it handy for your exam preparation

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