NCERT Solutions for 12th Class Maths: Chapter 9-Differential Equations

Class 12: Maths Chapter 9 solutions. Complete Class 12 Maths Chapter 9 Notes.

NCERT Solutions for 12th Class Maths: Chapter 9-Differential Equations

Class 12: Maths Chapter 9 solutions. Complete Class 12 Maths Chapter 9 Notes.

Ex 9.1 Class 12 Maths Question 1.

Solution:
Order of the equation is 4
It is not a polynomial in derivatives so that it
has not degree.

Ex 9.1 Class 12 Maths Question 2.

Solution:

It is a D.E. of order one and degree one.

Ex 9.1 Class 12 Maths Question 3.

Solution:
Order of the equation is 2.
Degree of the equation is

Ex 9.1 Class 12 Maths Question 4.

Solution:

It is a D.E. of order 2 and degree undefined

Ex 9.1 Class 12 Maths Question 5.

Solution:

It is a D.E. of order 2 and degree 1.

Ex 9.1 Class 12 Maths Question 6.

Solution:
Order of the equation is 3
Degree of the equation is 2

Ex 9.1 Class 12 Maths Question 7.

Solution:

The highest order derivative is y.
Thus the order of the D.E. is 3.
The degree of D.E is 1

Ex 9.1 Class 12 Maths Question 8.

Solution:

The order of the D. E. = 1 (highest order derivative)
The degree of the D.E. = 1.

Ex 9.1 Class 12 Maths Question 9.

Solution:

The highest derivative is 2.
Order of the D.E. = 2.
Degree of the D. E = 1

Ex 9.1 Class 12 Maths Question 10.

Solution:
Order of the equation is 2
Degree of the equation is 1

Ex 9.1 Class 12 Maths Question 11.
The degree of the differential equation

(a) 3
(b) 2
(c) 1
(d) not defined
Solution:

The degree not defined.
Because the differential equation can not be written as a polynomial in all the differential coefficients.
Hence option (d) is correct.

Ex 9.1 Class 12 Maths Question 12.
The order of the differential equation

(a) 2
(b) 1
(c) 0
(d) not defined
Solution:

Thus order of the D.E. = 2
Hence option (a) is correct.

Ex 9.2 Class 12 Maths Question 1.

Solution:

Ex 9.2 Class 12 Maths Question 2.

Solution:

Ex 9.2 Class 12 Maths Question 3.

Solution:

Ex 9.2 Class 12 Maths Question 4.

Solution:

Ex 9.2 Class 12 Maths Question 5.

Solution:

Ex 9.2 Class 12 Maths Question 6.

Solution:

Ex 9.2 Class 12 Maths Question 7.
xy = logy + C,

Solution:
xy = logy + C,

Ex 9.2 Class 12 Maths Question 8.

Solution:

Ex 9.2 Class 12 Maths Question 9.

Solution:

Ex 9.2 Class 12 Maths Question 10.

Solution:

Ex 9.2 Class 12 Maths Question 11.
The number of arbitrary constants in the general solution of a differential equation of fourth order are:
(a) 0
(b) 2
(c) 3
(d) 4
Solution:
(b) The general solution of a differential equation of fourth order has 4 arbitrary constants.
Because it contains the same number of arbitrary constants as the order of differential equation.

Ex 9.2 Class 12 Maths Question 12.
The number of arbitrary constants in the particular solution of a differential equation of third order are:
(a) 3
(b) 2
(c) 1
(d) 0
Solution:
(d) Number of arbitrary constants = 0
Because particular solution is free from arbitrary constants.

Ex 9.3 Class 12 Maths Question 1.

Solution:
Given that 

 …(i)
differentiating (i) w.r.t x, we get

 …(ii)
again differentiating w.r.t x, we get

which is the required differential equation

Ex 9.3 Class 12 Maths Question 2.
y² = a(b² – x²)
Solution:
given that
y² = a(b² – x²)…(i)

Ex 9.3 Class 12 Maths Question 3.
y = ae^3x+be^-2x
Solution:
Given that
y = ae^3x+be^-2x …(i)

Ex 9.3 Class 12 Maths Question 4.
y = e^2x (a+bx)
Solution:
y = e^2x (a+bx)

Ex 9.3 Class 12 Maths Question 5.
y = e^x(a cosx+b sinx)
Solution:
The curve y = e^x(a cosx+b sinx) …(i)
differentiating w.r.t x

Ex 9.3 Class 12 Maths Question 6.
Form the differential equation of the family of circles touching the y axis at origin
Solution:
The equation of the circle with centre (a, 0) and radius a, which touches y- axis at origin

Ex 9.3 Class 12 Maths Question 7.
Form the differential equation of the family of parabolas having vertex at origin and axis along positive y-axis.
Solution:
The equation of parabola having vertex at the origin and axis along positive y-axis is

Ex 9.3 Class 12 Maths Question 8.
Form the differential equation of family of ellipses having foci on y-axis and centre at origin.
Solution:
The equation of family ellipses having foci at y- axis is

Ex 9.3 Class 12 Maths Question 9.
Form the differential equation of the family of hyperbolas having foci on x-axis and centre at the origin.
Solution:
Equation of the hyperbola is 

Differentiating both sides w.r.t x

which is the req. differential eq. of the hyperbola.

Ex 9.3 Class 12 Maths Question 10.
Form the differential equation of the family of circles having centre on y-axis and radius 3 units
Solution:
Let centre be (0, a) and r = 3
Equation of circle is
x² + (y – a)² = 9 …(i)
Differentiating both sides, we get

which is required equation

Ex 9.3 Class 12 Maths Question 11.
Which of the following differential equation has 

 as the general solution ?
(a) 

(b) 

(c) 

(d) 

Solution:
(b) 

Ex 9.3 Class 12 Maths Question 12.
Which of the following differential equations has y = x as one of its particular solution ?
(a) 

(b) 

(c) 

(d) 

Solution:
(c) y = x

Ex 9.4 Class 12 Maths Question 1.

Solution:

integrating both sides, we get

Ex 9.4 Class 12 Maths Question 3.

Solution:

which is required solution

Ex 9.4 Class 12 Maths Question 4.
sec² x tany dx+sec² y tanx dy = 0
Solution:
we have
sec² x tany dx+sec² y tanx dy = 0

Ex 9.4 Class 12 Maths Question 5.

Solution:
we have

Integrating on both sides

Ex 9.4 Class 12 Maths Question 6.

Solution:

integrating on both side we get

which is required solution

Ex 9.4 Class 12 Maths Question 7.
y logy dx – x dy = 0
Solution:

integrating we get

Ex 9.4 Class 12 Maths Question 8.

Solution:

Ex 9.4 Class 12 Maths Question 9.
solve the following

Solution:

integrating both sides we get

Ex 9.4 Class 12 Maths Question 10.

Solution:

we can write in another form

Find a particular solution satisfying the given condition for the following differential equation in Q.11 to 14.

Ex 9.4 Class 12 Maths Question 11.

Solution:
here

integrating we get

Ex 9.4 Class 12 Maths Question 12.

Solution:

Ex 9.4 Class 12 Maths Question 13.

Solution:

Ex 9.4 Class 12 Maths Question 14.

Solution:

=> logy = logsecx + C
When x = 0, y = 1
=> log1 = log sec0 + C => 0 = log1 + C
=> C = 0
∴ logy = log sec x
=> y = sec x.

Ex 9.4 Class 12 Maths Question 15.
Find the equation of the curve passing through the point (0,0) and whose differential equation 

Solution:

Ex 9.4 Class 12 Maths Question 16.
For the differential equation 

 find the solution curve passing through the point (1,-1)
Solution:
The differential equation is

or xydy=(x + 2)(y+2)dx

Ex 9.4 Class 12 Maths Question 17.
Find the equation of a curve passing through the point (0, -2) given that at any point (pc, 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
Solution:
According to the question 

0, – 2) lies on it.c = 2
∴ Equation of the curve is : x² – y² + 4 = 0.

Ex 9.4 Class 12 Maths Question 18.
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).
Solution:
Slope of the tangent to the curve = 

slope of the line joining (x, y) and (- 4, – 3)

Ex 9.4 Class 12 Maths Question 19.
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and offer 3 seconds it is 6 units. Find the radius of balloon after t seconds.
Solution:
Let v be volume of the balloon.

Ex 9.4 Class 12 Maths Question 20.
In a bank principal increases at the rate of r% per year. Find the value of r if Rs 100 double itself in 10 years
Solution:
Let P be the principal at any time t.
According to the problem

Ex 9.4 Class 12 Maths Question 21.
In a bank principal increases 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
Solution:
Let p be the principal Rate of interest is 5%

Ex 9.4 Class 12 Maths Question 22.
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
Solution:
Let y denote the number of bacteria at any instant t • then according to the question

Ex 9.4 Class 12 Maths Question 23.
The general solution of a differential equation 

 is
(a) 

(b) 

(c) 

(d) 

Solution:
(a) 

Ex 9.5 Class 12 Maths Question 1.
(x²+xy)dy = (x²+y²)dx
Solution:
(x²+xy)dy = (x²+y²)dx

Ex 9.5 Class 12 Maths Question 2.

Solution:

Ex 9.5 Class 12 Maths Question 3.
(x-y)dy-(x+y)dx=0
Solution:

Ex 9.5 Class 12 Maths Question 4.
(x²-y²)dx+2xy dy=0
Solution:

Ex 9.5 Class 12 Maths Question 5.

Solution:

Ex 9.5 Class 12 Maths Question 6.

Solution:

Ex 9.5 Class 12 Maths Question 7.

Solution:

Ex 9.5 Class 12 Maths Question 8.

Solution:

Ex 9.5 Class 12 Maths Question 9.

Solution:

Ex 9.5 Class 12 Maths Question 10.

Solution:

For each of the following differential equation in Q 11 to 15 find the particular solution satisfying the given condition:

Ex 9.5 Class 12 Maths Question 11.
(x + y) dy+(x – y)dx = 0,y = 1 when x = 1
Solution:
given
(x + y) dy+(x – y)dx = 0

Ex 9.5 Class 12 Maths Question 12.
x²dy+(xy+y²)dx=0, y=1 when x=1
Solution:

f(x,y) is homogeneous
∴ put y = vx

Ex 9.5 Class 12 Maths Question 13.

Solution:

Ex 9.5 Class 12 Maths Question 14.

Solution:

which is a homogeneous differential equation

Ex 9.5 Class 12 Maths Question 15.

Solution:

 …(i)

Ex 9.5 Class 12 Maths Question 16.
A homogeneous equation of the form 

 can be solved by making the substitution,
(a) y=vx
(b) v=yx
(c) x=vy
(d) x=v
Solution:
(c) option x = vy

Ex 9.5 Class 12 Maths Question 17.
Which of the following is a homogeneous differential equation?
(a) (a) (4x + 6y + 5)dy-(3y + 2x + 4)dx = 0
(b) 

(c) 

(d) 

Solution:
(d)

Ex 9.6 Class 12 Maths Question 1.

Solution:
Given equation is a linear differential equation of the form 

;
Here, P = 2, Q = sin x

Ex 9.6 Class 12 Maths Question 2.

Solution:

Here P = 3, 

which is required equation

Ex 9.6 Class 12 Maths Question 3.

Solution:

Ex 9.6 Class 12 Maths Question 4.

Solution:
Here, P = secx, Q = tanx; 

= sec x + tan x
i.e., The solu. is y.× I.F. = ∫Q × I.F. dx + c
or y × (secx+tanx) = ∫tanx(secx+tanx)dx+c
Reqd. sol. is
∴ y(secx + tanx) = (secx + tanx)-x + c

Ex 9.6 Class 12 Maths Question 5.

Solution:

⇒ integrating factor = 

Ex 9.6 Class 12 Maths Question 6.

Solution:

Here P = 

 and Q = x logx

Ex 9.6 Class 12 Maths Question 7.

Solution:

Ex 9.6 Class 12 Maths Question 8.
(1+x²)dy+2xy dx = cotx dx(x≠0)
Solution:
(1+x²)dy+2xy dx = cotx dx

Ex 9.6 Class 12 Maths Question 9.

Solution:

Ex 9.6 Class 12 Maths Question 10.

Solution:

Ex 9.6 Class 12 Maths Question 11.

Solution:

Ex 9.6 Class 12 Maths Question 12.

Solution:

For each of the following Questions 13 to is find a particular solution, satisfying the given condition:

Ex 9.6 Class 12 Maths Question 13.

Solution:

Ex 9.6 Class 12 Maths Question 14.

Solution:

Ex 9.6 Class 12 Maths Question 15.

Solution:
Here P = -3cot x
Q = sin 2x

Ex 9.6 Class 12 Maths Question 16.
Find the equation of the 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
Solution:

Ex 9.6 Class 12 Maths Question 17.
Find the equation of the 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
Solution:
By the given condition

Ex 9.6 Class 12 Maths Question 18.
The integrating factor of the differential equation 

(a) 

(b) 

(c) 

(d) x
Solution:
(c) 

Ex 9.6 Class 12 Maths Question 19.
The integrating factor of the differential equation 

(-1<y<1) is
(a) 

(b) 

(c) 

(d) 

Solution:
(d) 

NCERT Solutions for 12th Class Maths: Chapter 9: Download PDF

NCERT Solutions for 12th Class Maths: Chapter 9-Differential Equations

Download PDF: NCERT Solutions for 12th Class Maths: Chapter 9-Differential Equations PDF

Chapterwise NCERT Solutions for Class 12 Maths :

• Chapter 1 – Relations and Functions
• Chapter 2 – Inverse Trigonometric Functions.
• Chapter 3 – Matrices
• Chapter 4 – Determinants.
• Chapter 5 – Continuity and Differentiability.0.0
• Chapter 6 – Application of Derivatives.
• Chapter 7 – Integrals.
• Chapter 8 – Application of Integrals.
• Chapter 9: Differential Equations
• Chapter 10: Vector Algebra
• Chapter 11: Three Dimensional Geometry
• Chapter 12: Linear Programming
• Chapter 13: Probability

About NCERT

The National Council of Educational Research and Training is an autonomous organization of the Government of India which was established in 1961 as a literary, scientific, and charitable Society under the Societies Registration Act. Its headquarters are located at Sri Aurbindo Marg in New Delhi. Visit the Official NCERT website to learn more.