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2: Inverse Trigonometric Functions
3: Matrices
4: Determinants
5: Continuity And Differentiability
6: Application Of Derivatives
7: Integrals
8: Application Of Integrals
▶ 9: Differential Equations
10: Vector Algebra
11: Three Dimensional Geometry
12: Linear Programming
13: Probability
![NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 9 - Differential Equations - Shaalaa.com](/images/mathematics-english-class-12_6:5f2b1b2038084cf381bfa42c826a928c.jpg "NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 9 - Differential Equations")
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Solutions for Chapter 9: Differential Equations
Below listed, you can find solutions for Chapter 9 of CBSE NCERT Exemplar for Mathematics [English] Class 12.
NCERT Exemplar solutions for Mathematics [English] Class 12 9 Differential Equations Solved Examples [Pages 180 - 192]
Short Answer
Find the differential equation of the family of curves y = Ae2x + B.e–2x.
Find the general solution of the differential equation `"dy"/"dx" = y/x`.
Given that `"dy"/"dx"` = yex and x = 0, y = e. Find the value of y when x = 1.
Solve the differential equation `"dy"/"dx" + y/x` = x2.
Find the differential equation of the family of lines through the origin.
Find the differential equation of all non-horizontal lines in a plane.
Find the equation of a curve whose tangent at any point on it, different from origin, has slope `y + y/x`.
Long Answer
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.
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`.
Solve `x^2 "dy"/"dx" - xy = 1 + cos(y/x)`, x ≠ 0 and x = 1, y = `pi/2`
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
The degree of the differential equation `(1 + "dy"/"dx")^3 = (("d"^2y)/("d"x^2))^2` is ______.
1
2
3
4
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
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
The order of the differential equation of all circles of given radius a is ______.
1
2
3
4
The solution of the differential equation `2x * "dy"/"dx" y` = 3 represents a family of ______.
Straight lines
Circles
Parabolas
Ellipses
The integrating factor of the differential equation `"dy"/"dx" (x log x) + y` = 2logx is ______.
ex
log x
log (log x)
x
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
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
Solution of the differential equation `"dx"/x + "dy"/y` = 0 is ______.
`1/x + 1/y` = c
logx . logy = c
xy = c
x + y = c
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:
Order of the differential equation representing the family of parabolas y2 = 4ax is ______.
The degree of the differential equation `("dy"/"dx")^2 + (("d"^2y)/("d"x^2))^2` = 0 is ______.
The number of arbitrary constants in a particular solution of the differential equation tan x dx + tan y dy = 0 is ______.
F(x, y) = `(sqrt(x^2 + y^2) + y)/x` is a homogeneous function of degree ______.
An appropriate substitution to solve the differential equation `"dx"/"dy" = (x^2 log(x/y) - x^2)/(xy log(x/y))` is ______.
Integrating factor of the differential equation `x "dy"/"dx" - y` = sinx is ______.
The general solution of the differential equation `"dy"/"dx" = "e"^(x - y)` is ______.
The general solution of the differential equation `"dy"/"dx" + y/x` = 1 is ______.
The differential equation representing the family of curves y = A sinx + B cosx is ______.
`("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
Order of the differential equation representing the family of ellipses having centre at origin and foci on x-axis is two.
True
False
Degree of the differential equation `sqrt(1 + ("d"^2y)/("d"x^2)) = x + "dy"/"dx"` is not defined.
True
False
`"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
F(x, y) = `(ycos(y/x) + x)/(xcos(y/x))` is not a homogeneous function.
True
False
F(x, y) = `(x^2 + y^2)/(x - y)` is a homogeneous function of degree 1.
True
False
Integrating factor of the differential equation `"dy"/"dx" - y` = cos x is ex.
True
False
The general solution of the differential equation x(1 + y2)dx + y(1 + x2)dy = 0 is (1 + x2)(1 + y2) = k.
True
False
The general solution of the differential equation `"dy"/"dx" + y sec x` = tan x is y(secx – tanx) = secx – tanx + x + k.
True
False
x + y = tan–1y is a solution of the differential equation `y^2 "dy"/"dx" + y^2 + 1` = 0.
True
False
y = x is a particular solution of the differential equation `("d"^2y)/("d"x^2) - x^2 "dy"/"dx" + xy` = x.
True
False
NCERT Exemplar solutions for Mathematics [English] Class 12 9 Differential Equations Exercise [Pages 193 - 203]
Short Answer
Find the solution of `"dy"/"dx"` = 2y–x.
Find the differential equation of all non-vertical lines in a plane.
Given that `"dy"/"dx" = "e"^-2x` and y = 0 when x = 5. Find the value of x when y = 3.
Solve the differential equation `(x^2 - 1) "dy"/"dx" + 2xy = 1/(x^2 - 1)`.
Solve the differential equation `"dy"/"dx" + 2xy` = y
Find the general solution of `"dy"/"dx" + "a"y` = emx
Solve the differential equation `"dy"/"dx" + 1` = ex + y.
Solve: ydx – xdy = x2ydx.
Solve the differential equation `"dy"/"dx"` = 1 + x + y2 + xy2, when y = 0, x = 0.
Find the general solution of `(x + 2y^3) "dy"/"dx"` = y
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)`.
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`.
Form the differential equation having y = (sin–1x)2 + Acos–1x + B, where A and B are arbitrary constants, as its general solution.
Form the differential equation of all circles which pass through origin and whose centres lie on y-axis.
Find the equation of a curve passing through origin and satisfying the differential equation `(1 + x^2) "dy"/"dx" + 2xy` = 4x2
Solve : `x^2 "dy"/"dx"` = x2 + xy + y2.
Find the general solution of the differential equation `(1 + y^2) + (x - "e"^(tan - 1y)) "dy"/"dx"` = 0.
Find the general solution of y2dx + (x2 – xy + y2) dy = 0.
Solve: (x + y)(dx – dy) = dx + dy. [Hint: Substitute x + y = z after seperating dx and dy]
Solve: `2(y + 3) - xy "dy"/"dx"` = 0, given that y(1) = – 2.
Solve the differential equation dy = cosx(2 – y cosecx) dx given that y = 2 when x = `pi/2`
Form the differential equation by eliminating A and B in Ax2 + By2 = 1
Solve the differential equation (1 + y2) tan–1xdx + 2y(1 + x2)dy = 0.
Find the differential equation of system of concentric circles with centre (1, 2).
Long Answer
Solve: `y + "d"/("d"x) (xy) = x(sinx + logx)`
Find the general solution of (1 + tany)(dx – dy) + 2xdy = 0.
Solve: `("d"y)/("d"x) = cos(x + y) + sin(x + y)`. [Hint: Substitute x + y = z]
Find the general solution of `("d"y)/("d"x) -3y = sin2x`
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)`.
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)`
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.
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.
Solcve: `x ("d"y)/("d"x) = y(log y – log x + 1)`
Objective Type from 34 to 75
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
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
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
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
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
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
Integrating factor of the differential equation `cosx ("d"y)/("d"x) + ysinx` = 1 is ______.
cosx
tanx
secx
sinx
Solution of the differential equation tany sec2xdx + tanx sec2ydy = 0 is ______.
tanx + tany = k
tanx – tany = k
`tanx/tany` = k
tanx . tany = k
Family y = Ax + A3 of curves is represented by the differential equation of degree ______.
1
2
3
4
Integrating factor of `(x"d"y)/("d"x) - y = x^4 - 3x` is ______.
x
logx
`1/x`
– x
Solution of `("d"y)/("d"x) - y` = 1, y(0) = 1 is given by ______.
xy = – ex
xy = – e-x
xy = – 1
y = 2ex – 1
The number of solutions of `("d"y)/("d"x) = (y + 1)/(x - 1)` when y (1) = 2 is ______.
None
One
Two
Infinite
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
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)`
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
The differential equation `y ("d"y)/("d"x) + "c"` represents: ______.
Family of hyperbolas
Family of parabolas
Family of ellipses
Family of circles
The general solution of ex cosy dx – ex siny dy = 0 is ______.
ex cosy = k
ex siny = k
ex = k cosy
ex = k siny
The degree of the differential equation `("d"^2y)/("d"x^2) + (("d"y)/("d"x))^3 + 6y^5` = 0 is ______.
1
2
3
5
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
Integrating factor of the differential equation `("d"y)/("d"x) + y tanx - secx` = 0 is ______.
cosx
secx
ecosx
esecx
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
The integrating factor of the differential equation `("d"y)/("d"x) + y = (1 + y)/x` is ______.
`x/"e"^x`
`"e"^x/x`
xex
ex
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
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
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`
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
Family y = Ax + A3 of curves will correspond to a differential equation of order ______.
3
2
1
Not defined
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
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
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`
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
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
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
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
The order and degree of the differential equation `[1 + ((dy)/(dx))^2] = (d^2y)/(dx^2)` are ______.
`2, 3/2`
2, 3
2, 1
3, 4
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`
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
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
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
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"`
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"`
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)
The degree of the differential equation `("d"^2y)/("d"x^2) + "e"^((dy)/(dx))` = 0 is ______.
The degree of the differential equation `sqrt(1 + (("d"y)/("d"x))^2)` = x is ______.
The number of arbitrary constants in the general solution of a differential equation of order three is ______.
`("d"y)/("d"x) + y/(xlogx) = 1/x` is an equation of the type ______.
General solution of the differential equation of the type `("d"x)/("d"x) + "P"_1x = "Q"_1` is given by ______.
The solution of the differential equation `x(dy)/("d"x) + 2y = x^2` is ______.
The solution of `(1 + x^2) ("d"y)/("d"x) + 2xy - 4x^2` = 0 is ______.
The solution of the differential equation ydx + (x + xy)dy = 0 is ______.
General solution of `("d"y)/("d"x) + y` = sinx is ______.
The solution of differential equation coty dx = xdy is ______.
The integrating factor of `("d"y)/("d"x) + y = (1 + y)/x` is ______.
State True or False for the following:
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
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
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
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
Number of arbitrary constants in the particular solution of a differential equation of order two is two.
True
False
The differential equation representing the family of circles x2 + (y – a)2 = a2 will be of order two.
True
False
The solution of `("d"y)/("d"x) = (y/x)^(1/3)` is `y^(2/3) - x^(2/3)` = c.
True
False
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
The solution of the differential equation `("d"y)/("d"x) = (x + 2y)/x` is x + y = kx2.
True
False
Solution of `x("d"y)/("d"x) = y + x tan y/x` is `sin(y/x)` = cx
True
False
The differential equation of all non horizontal lines in a plane is `("d"^2x)/("d"y^2)` = 0
True
False
Solutions for 9: Differential Equations
NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 9 - Differential Equations
Shaalaa.com has the CBSE Mathematics Mathematics [English] Class 12 CBSE solutions in a manner that help students grasp basic concepts better and faster. The detailed, step-by-step solutions will help you understand the concepts better and clarify any confusion. NCERT Exemplar solutions for Mathematics Mathematics [English] Class 12 CBSE 9 (Differential Equations) include all questions with answers and detailed explanations. This will clear students' doubts about questions and improve their application skills while preparing for board exams.
Further, we at Shaalaa.com provide such solutions so students can prepare for written exams. NCERT Exemplar textbook solutions can be a core help for self-study and provide excellent self-help guidance for students.
Concepts covered in Mathematics [English] 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, Differential Equations.
Using NCERT Exemplar Mathematics [English] Class 12 solutions Differential Equations exercise by students is an easy way to prepare for the exams, as they involve solutions arranged chapter-wise and also page-wise. The questions involved in NCERT Exemplar Solutions are essential questions that can be asked in the final exam. Maximum CBSE Mathematics [English] Class 12 students prefer NCERT Exemplar Textbook Solutions to score more in exams.
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