NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations: In class 11th, you have already learned how to differentiate a given function (f) with respect to an independent variable. In this article, you will get NCERT solutions for class 12 maths chapter 9 for all major topics of NCERT Class 12 maths syllabus. The equation of function and its one or more derivatives is called a differential equation.
In this chapter 9 NCERT Class 12 Maths solutions, some basic concepts related to the differential equations solutions, particular solutions, and general solutions of differential equations class 12 will be comprehensively discussed. In NCERT solutions for chapter 9 class 12 maths, questions from all these topics are covered in this article. If you are interested in other subjects then you can refer to NCERT solutions for class 12
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You will also learn some methods to find the differential equations solutions, the formation of differential equations class 12, and applications of differential equations in different areas in this NCERT Class 12 Maths chapter 9 solutions are also explained in details. Questions related to these topics are also covered in the NCERT solutions for class 12 maths ch 9 differential equations article. You can refer to NCERT solutions from classes 6 to 12 to learn CBSE maths and science.
Also read :
Class 12 Maths Chapter 9 Differential Equations Notes
NCERT Exemplar Solutions for Class 12 Maths Chapter 9 Differential Equations
NCERT Solutions for Class 12 Maths Chapter 9  Exercisewise Solved Questions
NCERT solutions for class 12 maths chapter 9 differential equationsExercise: 9.1
Question:1 Determine order and degree (if defined) of differential equation
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, the order of the given differential equation is 4
Now, the given differential equation is not a polynomial equation in its derivatives
Therefore, it's a degree is not defined
Question:2 Determine order and degree (if defined) of differential equation
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, the order of the given differential equation is 1
Now, the given differential equation is a polynomial equation in its derivatives and its highest power raised to y ' is 1
Therefore, it's a degree is 1.
Question:3 Determine order and degree (if defined) of differential equation
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, the order of the given differential equation is 2
Now, the given differential equation is a polynomial equation in its derivatives and power raised to s '' is 1
Therefore, it's a degree is 1
Question:4 Determine order and degree (if defined) of differential equation.
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, the order of the given differential equation is 2
Now, the given differential equation is not a polynomial equation in its derivatives
Therefore, it's a degree is not defined
Question:5 Determine order and degree (if defined) of differential equation.
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 2
Now, the given differential equation is a polynomial equation in it's dervatives and power raised to is 1
Therefore, it's degree is 1
Question:6 Determine order and degree (if defined) of differential equation
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 3 Now, the given differential equation is a polynomial equation in it's dervatives and power raised to is 2
Therefore, it's degree is 2
Question:7 Determine order and degree (if defined) of differential equation
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 3
Now, the given differential equation is a polynomial equation in it's dervatives and power raised to is 1
Therefore, it's degree is 1
Question:8 Determine order and degree (if defined) of differential equation
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 1
Now, the given differential equation is a polynomial equation in it's dervatives and power raised to is 1
Therefore, it's degree is 1
Question:9 Determine order and degree (if defined) of differential equation
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 2
Now, the given differential equation is a polynomial equation in it's dervatives and power raised to is 1
Therefore, it's degree is 1
Question:10 Determine order and degree (if defined) of differential equation
Answer:
Given function is
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 2
Now, the given differential equation is a polynomial equation in it's dervatives and power raised to is 1
Therefore, it's degree is 1
Question:11 The degree of the differential equation is
(A) 3
(B) 2
(C) 1
(D) not defined
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 2
Now, the given differential equation is a not polynomial equation in it's dervatives
Therefore, it's degree is not defined
Therefore, answer is (D)
Question:12 The order of the differential equation is
(A) 2
(B) 1
(C) 0
(D) Not Defined
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, order of given differential equation is 2
Therefore, answer is (A)
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NCERT solutions for class 12 maths chapter 9 differential equationsExercise: 9.2
Question:1 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Again, differentiating both sides w.r.t. x,
Substituting the values of y’ and y'' in the given differential equations,
y''  y' = e ^{ x }  e ^{ x } = 0 = RHS.
Therefore, the given function is the solution of the corresponding differential equation.
Question:2 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y’ in the given differential equations,
.
Therefore, the given function is the solution of the corresponding differential equation.
Question:3. Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y’ in the given differential equations,
.
Therefore, the given function is not the solution of the corresponding differential equation.
Question:4. Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y in RHS,
.
Therefore, the given function is a solution of the corresponding differential equation.
Question:5 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y' in LHS,
.
Therefore, the given function is a solution of the corresponding differential equation.
Question:6. Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y' in LHS,
Substituting the values of y in RHS.
Therefore, the given function is a solution of the corresponding differential equation.
Question:7 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y' in LHS,
Therefore, the given function is a solution of the corresponding differential equation.
Question:8 In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
y' + siny.y' = 1
y'(1 + siny) = 1
Substituting the values of y and y' in LHS,
= (x + cosy) = y = RHS
Therefore, the given function is a solution of the corresponding differential equation.
Question:9 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y' in LHS,
Therefore, the given function is a solution of the corresponding differential equation.
Question:10 Verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation:
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Substituting the values of y and y' in LHS,
Therefore, the given function is a solution of the corresponding differential equation.
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
Answer:
(D) 4
The number of constants in the general solution of a differential equation of order n is equal to its order.
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
Answer:
(D) 0
In a particular solution of a differential equation, there is no arbitrary constant.
NCERT solutions for class 12 maths chapter 9 differential equationsExercise: 9.3
Question:1 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
Answer:
Given equation is
Differentiate both the sides w.r.t x
Now, again differentiate it w.r.t x
Therefore, the required differential equation is or
Question:2 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
Answer:
Given equation is
Differentiate both the sides w.r.t x
(i)
Now, again differentiate it w.r.t x
(ii)
Now, divide equation (i) and (ii)
Therefore, the required differential equation is
Question:3 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
Answer:
Given equation is
(i)
Differentiate both the sides w.r.t x
(ii)
Now, again differentiate w.r.t. x
(iii)
Now, multiply equation (i) with 2 and add equation (ii)
(iv)
Now, multiply equation (i) with 3 and subtract from equation (ii)
(v)
Now, put values from (iv) and (v) in equation (iii)
Therefore, the required differential equation is
Question:4 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
Answer:
Given equation is
(i)
Now, differentiate w.r.t x
(ii)
Now, again differentiate w.r.t x
(iii)
Now, multiply equation (ii) with 2 and subtract from equation (iii)
(iv)
Now,put the value in equation (iii)
Therefore, the required equation is
Question:5 Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b.
Answer:
Given equation is
(i)
Now, differentiate w.r.t x
(ii)
Now, again differentiate w.r.t x
(iii)
Now, multiply equation (i) with 2 and multiply equation (ii) with 2 and add and subtract from equation (iii) respectively
we will get
Therefore, the required equation is
Question:6 Form the differential equation of the family of circles touching the yaxis at origin.
Answer:
If the circle touches yaxis at the origin then the centre of the circle lies at the xaxis
Let r be the radius of the circle
Then, the equation of a circle with centre at (r,0) is
(i)
Now, differentiate w.r.t x
(ii)
Put equation (ii) in equation (i)
Therefore, the required equation is
Question:7 Form the differential equation of the family of parabolas having vertex at origin and axis along positive yaxis.
Answer:
Equation of perabola having vertex at origin and axis along positive yaxis is
(i)
Now, differentiate w.r.t. c
(ii)
Put value from equation (ii) in (i)
Therefore, the required equation is
Question:8 Form the differential equation of the family of ellipses having foci on yaxis and centre at origin.
Answer:
Equation of ellipses having foci on yaxis and centre at origin is

Now, differentiate w..r.t. x
(i)
Now, again differentiate w.r.t. x
(ii)
Put value from equation (ii) in (i)
Our equation becomes
Therefore, the required equation is
Question:9 Form the differential equation of the family of hyperbolas having foci on xaxis and centre at origin.
Answer:
Equation of hyperbolas having foci on xaxis and centre at the origin
Now, differentiate w..r.t. x
(i)
Now, again differentiate w.r.t. x
(ii)
Put value from equation (ii) in (i)
Our equation becomes
Therefore, the required equation is
Question:10 Form the differential equation of the family of circles having centre on yaxis and radius 3 units.
Answer:
Equation of the family of circles having centre on yaxis and radius 3 units
Let suppose centre is at (0,b)
Now, equation of circle with center (0,b) an radius = 3 units
Now, differentiate w.r.t x
we get,
Put value fro equation (ii) in (i)
Therefore, the required differential equation is
Question:11 Which of the following differential equations has as the general solution?
(A)
(B)
(C)
(D)
Answer:
Given general solution is
Differentiate it w.r.t x
we will get
Again, Differentiate it w.r.t x
Therefore, (B) is the correct answer
Question:12 Which of the following differential equations has as one of its particular solution?
(A)
(B)
(C)
(D)
Answer:
Given equation is
Now, on differentiating it w.r.t x
we get,
and again on differentiating it w.r.t x
we get,
Now, on substituting the values of in all the options we will find that only option c which is satisfies
Therefore, the correct answer is (C)
NCERT solutions for class 12 maths chapter 9 differential equationsExercise: 9.4
Question:1 Find the general solution:
Answer:
Given,
Question:2 Find the general solution:
Answer:
Given, in the question
The required general solution:
Question:3 Find the general solution:
Answer:
Given, in the question
The required general equation
Question:4 Find the general solution:
Answer:
Given,
Now, let tany = t and tanx = u
Question:5 Find the general solution:
Answer:
Given, in the question
Let,
This is the general solution
Question:6 Find the general solution:
Answer:
Given, in the question
Question:7 Find the general solution:
Answer:
Given,
let logy = t
=> 1/ydy = dt
This is the general solution
Question:8 Find the general solution:
Answer:
Given, in the question
This is the required general equation.
Question:9 Find the general solution:
Answer:
Given, in the question
Now,
Here, u = and v = 1
Question:10 Find the general solution
Answer:
Given,
Question:11 Find a particular solution satisfying the given condition:
Answer:
Given, in the question
Now,
Now comparing the coefficients
A + B = 2; B + C = 1; A + C = 0
Solving these:
Putting the values of A,B,C:
Therefore,
Now, y= 1 when x = 0
c = 1
Putting the value of c, we get:
Question:12 Find a particular solution satisfying the given condition:
Answer:
Given, in the question
Let,
Now comparing the values of A,B,C
A + B + C = 0; BC = 0; A = 1
Solving these:
Now putting the values of A,B,C
Given, y =0 when x =2
Therefore,
Question:13 Find a particular solution satisfying the given condition:
Answer:
Given,
Now, y =1 when x =0
1 = 0 + c
Therefore, c = 1
Putting the value of c:
Question:14 Find a particular solution satisfying the given condition:
Answer:
Given,
Now, y=1 when x =0
1 = ksec0
k = 1
Putting the vlue of k:
y = sec x
Question:15 Find the equation of a curve passing through the point (0, 0) and whose differential equation is .
Answer:
We first find the general solution of the given differential equation
Given,
Now, Since the curve passes through (0,0)
y = 0 when x =0
Putting the value of c, we get:
Question:16 For the differential equation , find the solution curve passing through the point (1, –1).
Answer:
We first find the general solution of the given differential equation
Given,
Now, Since the curve passes through (1,1)
y = 1 when x = 1
Putting the value of C:
Question:17 Find the equation of a curve passing through the point given that at any point 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.
Answer:
According to the question,
Now, Since the curve passes through (0,2).
x =0 and y = 2
Putting the value of c, we get
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).
Answer:
Slope m of line joining (x,y) and (4,3) is
According to the question,
Now, Since the curve passes through (2,1)
x = 2 , y =1
Putting the value of k, we get
Question:19 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.
Answer:
Volume of a sphere,
Given, Rate of change is constant.
Now, at t=0, r=3 and at t=3 , r =6
Putting these value:
Also,
Putting the value of c and k:
Question:20 In a bank, principal increases continuously at the rate of r % per year. Find the value of r if Rs 100 double itself in 10 years (log _{ e } 2 = 0.6931).
Answer:
Let p be the principal amount and t be the time.
According to question,
Now, at t =0 , p = 100
and at t =10, p = 200
Putting these values,
Also,
,
So value of r = 6.93%
Question:21 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 (e ^{ 0.5 } = 1.648).
Answer:
Let p be the principal amount and t be the time.
According to question,
Now, at t =0 , p = 1000
Putting these values,
Also, At t=10
,
After 10 years, the total amount would be Rs.1648
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?
Answer:
Let n be the number of bacteria at any time t.
According to question,
Now, at t=0, n = 100000
Again, at t=2, n= 110000
Using these values, for n= 200000
Question:23 The general solution of the differential equation is
(A)
(B)
(C)
(D)
Answer:
Given,
NCERT solutions for class 12 maths chapter 9 differential equationsExercise: 9.5
Question:1 Show that the given differential equation is homogeneous and solve each of them.
Answer:
The given diffrential eq can be written as
Let
Now,
Hence, it is a homogeneous equation.
To solve it put y = vx
Diff erentiating on both sides wrt
Substitute this value in equation (i)
Integrating on both side, we get;
Again substitute the value ,we get;
This is the required solution of given diff. equation
Question:2 Show that the given differential equation is homogeneousand solve each of them.
Answer:
the above differential eq can be written as,
............................(i)
Now,
Thus the given differential eq is a homogeneous equaion
Now, to solve substitute y = vx
Diff erentiating on both sides wrt
Substitute this value in equation (i)
Integrating on both sides, we get; (and substitute the value of )
this is the required solution
Question:3 Show that the given differential equation is homogeneous and solve each of them.
Answer:
The given differential eq can be written as;
....................................(i)
Hence it is a homogeneous equation.
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
Integrating on both sides, we get;
again substitute the value of
This is the required solution.
Question:4 Show that the given differential equation is homogeneous and solve each of them.
Answer:
we can write it as;
...................................(i)
Hence it is a homogeneous equation
Now, to solve substitute y = vx
Diff erentiating on both sides wrt
Substitute this value in equation (i)
integrating on both sides, we get
.............[ ]
This is the required solution.
Question:5 Show that the given differential equation is homogeneous and solve it.
Answer:
............(i)
Hence it is a homogeneous eq
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
On integrating both sides, we get;
after substituting the value of
This is the required solution
Question:6 Show that the given differential equation is homogeneous and solve it.
Answer:
.................................(i)
henxe it is a homogeneous equation
Now, to solve substitute y = vx
Diff erentiating on both sides wrt
Substitute this value in equation (i)
On integrating both sides,
Substitute the value of v=y/x , we get
Required solution
Question:7 Solve.
Answer:
......................(i)
By looking at the equation we can directly say that it is a homogenous equation.
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
integrating on both sides, we get
substitute the value of v= y/x , we get
Required solution
Question:8 Solve.
Answer:
...............................(i)
it is a homogeneous equation
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
On integrating both sides we get;
Required solution
Question:9 Solve.
Answer:
..................(i)
hence it is a homogeneous eq
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
integrating on both sides, we get; ( substituting v =y/x)
This is the required solution of the given differential eq
Question:10 Solve.
Answer:
.......................................(i)
Hence it is a homogeneous equation.
Now, to solve substitute x = yv
Diff erentiating on both sides wrt
Substitute this value in equation (i)
Integrating on both sides, we get;
This is the required solution of the diff equation.
Question:11 Solve for particular solution.
Answer:
..........................(i)
We can clearly say that it is a homogeneous equation.
Now, to solve substitute y = vx
Diff erentiating on both sides wrt
Substitute this value in equation (i)
On integrating both sides
......................(ii)
Now, y=1 and x= 1
After substituting the value of 2k in eq. (ii)
This is the required solution.
Question:12 Solve for particular solution.
Answer:
...............................(i)
Hence it is a homogeneous equation
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i), we get
Integrating on both sides, we get;
replace the value of v=y/x
.............................(ii)
Now y =1 and x = 1
therefore,
Required solution
Question:13 Solve for particular solution.
Answer:
..................(i)
Hence it is a homogeneous eq
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
on integrating both sides, we get;
On substituting v =y/x
............................(ii)
Now,
put this value of C in eq (ii)
Required solution.
Question:14 Solve for particular solution.
Answer:
....................................(i)
the above eq is homogeneous. So,
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
on integrating both sides, we get;
.................................(ii)
now y = 0 and x =1 , we get
put the value of C in eq 2
Question:15 Solve for particular solution.
Answer:
The above eq can be written as;
By looking, we can say that it is a homogeneous equation.
Now, to solve substitute y = vx
Differentiating on both sides wrt
Substitute this value in equation (i)
integrating on both sides, we get;
.............................(ii)
Now, y = 2 and x =1, we get
C =1
put this value in equation(ii)
Question:16 A homogeneous differential equation of the from can be solved by making the substitution.
(A)
(B)
(C)
(D)
Answer:
for solving this type of equation put x/y = v
x = vy
option C is correct
Question:17 Which of the following is a homogeneous differential equation?
(A)
(B)
(C)
(D)
Answer:
Option D is the right answer.
we can take out lambda as a common factor and it can be cancelled out
NCERT solutions for class 12 maths chapter 9 differential equationsExercise: 9.6
Question:1 Find the general solution:
Answer:
Given equation is
This is type where p = 2 and Q = sin x
Now,
Now, the solution of given differential equation is given by relation
Let
Put the value of I in our equation
Now, our equation become
Therefore, the general solution is
Question:2 Solve for general solution:
Answer:
Given equation is
This is type where p = 3 and
Now,
Now, the solution of given differential equation is given by the relation
Therefore, the general solution is
Question:3 Find the general solution
Answer:
Given equation is
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Therefore, the general solution is
Question:4 Solve for General Solution.
Answer:
Given equation is
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Therefore, the general solution is
Question:5 Find the general solution.
Answer:
Given equation is
we can rewrite it as
This is where and
Now,
Now, the solution of given differential equation is given by relation
take
Now put again
Put this value in our equation
Therefore, the general solution is
Question:6 Solve for General Solution.
Answer:
Given equation is
Wr can rewrite it as
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Let
Put this value in our equation
Therefore, the general solution is
Question:7 Solve for general solutions.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of given differential equation is given by relation
take
Put this value in our equation
Therefore, the general solution is
Question:8 Find the general solution.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of the given differential equation is given by the relation
Therefore, the general solution is
Question:9 Solve for general solution.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of the given differential equation is given by the relation
Lets take
Put this value in our equation
Therefore, the general solution is
Question:10 Find the general solution.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Lets take
Put this value in our equation
Therefore, the general solution is
Question:11 Solve for general solution.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Therefore, the general solution is
Question:12 Find the general solution.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Therefore, the general solution is
Question:13 Solve for particular solution.
Answer:
Given equation is
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Now, by using boundary conditions we will find the value of C
It is given that y = 0 when
at
Now,
Therefore, the particular solution is
Question:14 Solve for particular solution.
Answer:
Given equation is
we can rewrite it as
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Now, by using boundary conditions we will find the value of C
It is given that y = 0 when x = 1
at x = 1
Now,
Therefore, the particular solution is
Question:15 Find the particular solution.
Answer:
Given equation is
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Now, by using boundary conditions we will find the value of C
It is given that y = 2 when
at
Now,
Therefore, the particular solution is
Question:16 Find the equation of a 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.
Answer:
Let f(x , y) is the curve passing through origin
Then, the slope of tangent to the curve at point (x , y) is given by
Now, it is given that
It is type of equation where
Now,
Now,
Now, Let
Put this value in our equation
Now, by using boundary conditions we will find the value of C
It is given that curve passing through origin i.e. (x , y) = (0 , 0)
Our final equation becomes
Therefore, the required equation of the curve is
Question:17 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.
Answer:
Let f(x , y) is the curve passing through point (0 , 2)
Then, the slope of tangent to the curve at point (x , y) is given by
Now, it is given that
It is type of equation where
Now,
Now,
Now, Let
Put this value in our equation
Now, by using boundary conditions we will find the value of C
It is given that curve passing through point (0 , 2)
Our final equation becomes
Therefore, the required equation of curve is
Question:18 The Integrating Factor of the differential equation is
(A)
(B)
(C)
(D)
Answer:
Given equation is
we can rewrite it as
Now,
It is type of equation where
Now,
Therefore, the correct answer is (C)
Question:19 The Integrating Factor of the differential equation is
(A)
(B)
(C)
(D)
Answer:
Given equation is
we can rewrite it as
It is type of equation where
Now,
Therefore, the correct answer is (D)
NCERT solutions for class 12 maths chapter 9 differential equationsMiscellaneous Exercise
Question:1 Indicate Order and Degree.
(i)
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is
Therefore, the order of the given differential equation is 2
Now, the given differential equation is a polynomial equation in its derivative y '' and y 'and power raised to y '' is 1
Therefore, it's degree is 1
Question:1 Indicate Order and Degree.
(ii)
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is y'
Therefore, order of given differential equation is 1
Now, the given differential equation is a polynomial equation in it's dervatives y 'and power raised to y ' is 3
Therefore, it's degree is 3
Question:1 Indicate Order and Degree.
(iii)
Answer:
Given function is
We can rewrite it as
Now, it is clear from the above that, the highest order derivative present in differential equation is y''''
Therefore, order of given differential equation is 4
Now, the given differential equation is not a polynomial equation in it's dervatives
Therefore, it's degree is not defined
Question:2 Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i)
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Again, differentiating both sides w.r.t. x,
Therefore, the given function is the solution of the corresponding differential equation.
Question:2 Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(ii)
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Again, differentiating both sides w.r.t. x,
Therefore, the given function is the solution of the corresponding differential equation.
Question:2 Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(iii)
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Again, differentiating both sides w.r.t. x,
Therefore, the given function is the solution of the corresponding differential equation.
Question:2 Verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(iv)
Answer:
Given,
Now, differentiating both sides w.r.t. x,
Putting values in LHS
Therefore, the given function is the solution of the corresponding differential equation.
Question:3 Form the differential equation representing the family of curves given by , where a is an arbitrary constant.
Answer:
Given equation is
we can rewrite it as
(i)
Differentiate both the sides w.r.t x
(ii)
Put value from equation (ii) in (i)
Therefore, the required differential equation is
Question:4 Prove that is the general solution of differential equation , where c is a parameter.
Answer:
Given,
Now, let y = vx
Substituting the values of y and y' in the equation,
Integrating both sides we get,
Now,
Let
Now,
Let v ^{ 2 } = p
Now, substituting the values of I _{ 1 } and I _{ 2 } in the above equation, we get,
Thus,
Question:5 Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Answer:
Now, equation of the circle with center at (x,y) and radius r is
Since, it touch the coordinate axes in first quadrant
Therefore, x = y = r
(i)
Differentiate it w.r.t x
we will get
(ii)
Put value from equation (ii) in equation (i)
Therefore, the differential equation of the family of circles in the first quadrant which touches the coordinate axes is
Question:6 Find the general solution of the differential equation
Answer:
Given equation is
we can rewrite it as
Now, integrate on both the sides
Therefore, the general solution of the differential equation is
Question:7 Show that the general solution of the differential equation is given by , where A is parameter.
Answer:
Given,
Integrating both sides,
Let
Let A = ,
Hence proved.
Question:8 Find the equation of the curve passing through the point whose differential equation is
Answer:
Given equation is
we can rewrite it as
Integrate both the sides
Now by using boundary conditiond, we will find the value of C
It is given that the curve passing through the point
So,
Now,
Therefore, the equation of the curve passing through the point whose differential equation is is
Question:9 Find the particular solution of the differential equation , given that when .
Answer:
Given equation is
we can rewrite it as
Now, integrate both the sides
Put
Put again
Put this in our equation
Now, by using boundary conditions we will find the value of C
It is given that
y = 1 when x = 0
Now, put the value of C
Therefore, the particular solution of the differential equation is
Question:10 Solve the differential equation
Answer:
Given,
Let
Differentiating it w.r.t. y, we get,
Thus from these two equations,we get,
Question:11 Find a particular solution of the differential equation , given that , when . (Hint: put )
Answer:
Given equation is
Now, integrate both the sides
Put
Now, given equation become
Now, integrate both the sides
Put again
Now, by using boundary conditions we will find the value of C
It is given that
y = 1 when x = 0
Now, put the value of C
Therefore, the particular solution of the differential equation is
Question:12 Solve the differential equation .
Answer:
Given,
This is equation is in the form of
p = and Q =
Now, I.F. =
We know that the solution of the given differential equation is:
Question:13 Find a particular solution of the differential equation , given that .
Answer:
Given equation is
This is type where and
Now,
Now, the solution of given differential equation is given by relation
Now, by using boundary conditions we will find the value of C
It is given that y = 0 when
at
Now, put the value of C
Therefore, the particular solution is
Question:14 Find a particular solution of the differential equation , given that when
Answer:
Given equation is
we can rewrite it as
Integrate both the sides
Put
put again
Put this in our equation
Now, by using boundary conditions we will find the value of C
It is given that y = 0 when x = 0
at x = 0
Now, put the value of C
Therefore, the particular solution is
Question:15 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 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
Answer:
Let n be the population of the village at any time t.
According to question,
Now, at t=0, n = 20000 (Year 1999)
Again, at t=5, n= 25000 (Year 2004)
Using these values, at t =10 (Year 2009)
Therefore, the population of the village in 2009 will be 31250.
Question:16 The general solution of the differential equation is
(A)
(B)
(C)
(D)
Answer:
Given equation is
we can rewrite it as
Integrate both the sides
we will get
Therefore, answer is (C)
Question:17 The general solution of a differential equation of the type is
(A)
(B)
(C)
(D)
Answer:
Given equation is
and we know that the general equation of such type of differential equation is
Therefore, the correct answer is (C)
Question:18 The general solution of the differential equation is
(A)
(B)
(C)
(D)
Answer:
Given equation is
we can rewrite it as
It is type of equation where
Now,
Now, the general solution is
Therefore, (C) is the correct answer
If you want to get command on concepts then differential equations solutions of NCERT exercise are listed below
 Differential Equations Class 12 Exercise 9.1
 Differential Equations Class 12 Exercise 9.2
 Differential Equations Class 12 Exercise 9.3
 Differential Equations Class 12 Exercise 9.4
 Differential Equations Class 12 Exercise 9.5
 Differential Equations Class 12 Exercise 9.6
 Differential Equations Class 12 Miscellaneous Exercise
More About NCERT Solutions for Class 12 Maths Chapter 9
This Chapter Differential Equations Class 12 has 5 marks weightage in 12th board final examination. Generally, one question is asked from this Chapter 9 Class 12 Maths that can be studied in detail from the NCERT Class 12 maths book in the 12th board final exam. You can score these 5 marks very easily with the help of these Ncert Solutions For Class 12 Maths Chapter 9 Differential Equations.
Class 12 Maths ch 9 is very important for the students aspiring for the 12th board exam. This NCERT Class 12 Maths Chapter 9 solutions holds good weightage in competitive exams like JEE Main, VITEEE, BITSAT. In this chapter, there are 6 exercises with 95 questions. All these questions are prepared and explained in this NCERT solutions for class 12 maths chapter 9 differential equations article.
Topics of NCERT class 12 maths chapter 9 Differential Equations
9.1 Introduction
9.2 Basic Concepts
9.2.1. Order of a differential equation
9.2.2 Degree of a differential equation
9.3. General and Particular Solutions of a Differential Equation
9.4 Formation of a Differential Equation whose General Solution is given
9.4.1 Procedure to form a differential equation that will represent a given family of curves
9.5. Methods of Solving First Order, First Degree Differential Equations
9.5.1 Differential equations with variables separable
9.5.2 Homogeneous differential equations
9.5.3 Linear differential equations
Also, check NCERT Exemplar Class 12 Solutions
NCERT Exemplar Class 12th Maths Solutions
NCERT Exemplar Class 12th Physics Solutions
NCERT Exemplar Class 12th Chemistry Solutions
NCERT Exemplar Class 12th Biology Solutions
NCERT Exemplar Class 12 Maths Solutions Chapter 9
So, what is basically a differential equation? A differential equation is an equation in which derivatives of the dependent variable with respect to independent variables involved. Let's understand it with an example from NCERT chapter 9 differential equations
From the above equations, we notice that equations (1), (2) and (3) involve dependent variable(variables) and/or independent only but equation (4) involves variables as well as derivative of the dependent variable (y) with respect to the independent variable (x). That type of equation is known as the differential equation.
Important terms used in class 12 chapter 9 differential equations
 Order of a differential equation  It is the order of the highest order derivative present in the equation.
 Degree of a differential equation  It is the power of the highest order derivative in the differential equation.
 Homogeneous differential equation  A differential equation that can be expressed in the form where is a homogeneous function of degree zero.
 First order linear differential equation  A differential equation of the form where P and Q are constants or functions of x only.
NCERT solutions for class 12 maths  Chapterwise
chapter 1  NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions 
chapter 2  NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions 
chapter 3  NCERT solutions for class 12 maths chapter 3 Matrices 
chapter 4  NCERT solutions for class 12 maths chapter 4 Determinants 
chapter 5  NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability 
chapter 6  NCERT solutions for class 12 maths chapter 6 Application of Derivatives 
chapter 7  NCERT solutions for class 12 maths chapter 7 Integrals 
chapter 8  NCERT solutions for class 12 maths chapter 8 Application of Integrals 
chapter 9  NCERT solutions for class 12 maths chapter 9 Differential Equations 
chapter 10  NCERT solutions for class 12 maths chapter 10 Vector Algebra 
chapter 11  NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry 
chapter 12  NCERT solutions for class 12 maths chapter 12 Linear Programming 
chapter 13  NCERT solutions for class 12 maths chapter 13 Probability 
NCERT solutions for class 12 subject wise
 NCERT solutions for class 12 mathematics
 NCERT solutions class 12 chemistry
 NCERT solutions for class 12 physics
 NCERT solutions for class 12 biology
NCERT Solutions class wise
 NCERT solutions for class 12
 NCERT solutions for class 11
 NCERT solutions for class 10
 NCERT solutions for class 9
Tips to use NCERT Class 12 Maths Chapter 9 Solutions
NCERT solutions for class 12 maths chapter 9 differential equations are very helpful for the preparation of this chapter. Here are some tips to get command on it.
 Differential equations are the easiest part of the class 12 calculus. If your concepts of integration are clear then it won't take much effort to get command on this chapter.
 First, solve all NCERT problems including examples and miscellaneous exercise on your own. If you are not able to solve you can take the help of NCERT solutions for class 12 maths chapter 9 differential equations which are provided here
 If you have solved NCERT problems, you can solve previous years paper. It gives you an idea about the type of questions and difficulty levels of questions that have been asked in previous years
Happy learning !!!