NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise (Differential Equations)
NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise contains all the questions with detailed solutions. Students are advised to solve these questions for better understanding of the concepts.
Krati Saraswat2 Feb, 2024
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NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise (Differential Equations)
NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise Differential Equations is prepared by the academic team of Physics Wallah. We have prepared NCERT Solutions for all exercise of Chapter 9. Given below are step-by-step solutions to all questions given in the NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise.NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise Overview
NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise is prepared by our experts to help students understand the concepts of the chapter better. Students can solve these questions before their examinations these questions will help them to understand the concepts better and by doing these questions students can easily ace their examinations.NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise
Solve The Following Questions of NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise:
Question 1. For each of the differential equations given below, indicate its order and degree (if defined): (i)
(ii)
(iii)
Solution :
(i) Given: Differential equation
The highest order derivative present in this differential equation is d
2
y/dx
2
and hence order of this differential equation if 2.
The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative d
2
y/dx
2
is 1.
Therefore, Order = 2, Degree = 1
(ii) Given: Differential equation
The highest order derivative present in this differential equation is dy/dx and hence order of this differential equation if 1.
The given differential equation is a polynomial equation in derivatives and highest power of the highest order derivative dy/dx is 3.
Therefore, Order = 1, Degree = 3
(iii) Given: Differential equation
The highest order derivative present in this differential equation is d
4
y/dx
4
and hence order of this differential equation if 4.
The given differential equation is not a polynomial equation in derivatives therefore, degree of this differential equation is not defined.
Therefore, Order = 4, Degree not defined.
Solution :
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
Therefore, Function given by eq. (i) is a solution of D.E. (ii).
Question 3. Form the differential equation representing the family of curves
where a ia an arbitrary constant.
Solution :
Equation of the given family of curves is
Question 4. Prove that
is the general equation of the differential equation
where c is a parameter.
Solution :
Integrating both sides, we get:
Substituting the values of
I
1
and
I
2
in equation (3), we get:
Hence, the given result is proved.
Question 5. For the differential equation of the family of the circles in the first quadrant which touch the coordinate axes.
Solution :
The equation of a circle in the first quadrant with centre (
a
,
a
) and radius (
a)
which touches the coordinate axes is:
Question 6. Find the general solution of the differential equation
Solution :
Given: Differential Equation
Question 7. Show that the general solution of the differential equation
is given by (
x
+
y
+ 1) =
A
(1 –
x
–
y
– 2
xy
), where
A is parameter
Solution :
Given: Differential equation
Integrating both sides,
Question 8. Find the equation of the curve passing through the point (0,π/4), whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Solution :
The differential equation of the given curve is:
Question 9.Find the particular solution of the differential equation (1 + e
2x
) dy + (1 + y
2
) ex dx = 0, given that y = 1 when x = 0.
Solution :
This is the required particular solution of the given differential equation.
Question 10. Solve the differential equation:
Solution :
Question 11. Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)
Solution :
Now,
y
= –1 at
x
= 0.
Therefore, equation (3) becomes:
log 1 = 0 – 1 + C
⇒ C = 1
Substituting C = 1 in equation (3) we get:
log| x- y| = x + y + 1
This is the required particular solution of the given differential equati
on.
Question 12. Solve the differential equation:
Solution :
Question 13. Find the particular solution of the differential equation
given that
y
= 0 when x = π/2
Solution :
The given differential equation is:
This is the required particular solution of the given differential equation.
Question 14. Find the particular solution of the differential equation
given that
y
= 0 when
x
= 0
Solution :
Now, at
x
= 0 and
y
= 0, equation (2) becomes:
This is the required particular solution of the given differential equation.
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 25,000 in the year 2004, what will be the population of the village in 2009?
Solution :
Let the population at any instant (
t)
be
y
.
It is given that the rate of increase of population is proportional to the number of inhabitants at any instant.
Integrating both sides, we get:
log
y
=
kt
+ C … (1)
In the year 1999,
t
= 0 and
y
= 20000.
Therefore, we get:
log 20000 = C … (2)
In the year 2004,
t
= 5 and
y
= 25000.
Therefore, we get:
In the year 2009,
t
= 10 years.
Now, on substituting the values of
t
,
k,
and C in equation (1), we get:
Hence, the population of the village in 2009 will be 31250.
Choose the correct answer:
Question 16. The general solution of the differential equation
is:
(A)
xy
= C
(B)
x
= C
y
2
(C)
y
= C
x
(D)
y
= C
x
2
Solution :
The given differential equation is:
Therefore, option (C) is correct.
Question 17. The general equation of a differential equation of the type
is:
Solution :
The integrating factor of the given differential equation
The general solution of the differential equation is given by,
Hence, the correct answer is C
Question 18. The general solution of the differential equation
is:
(A)
xe
y
+
x
2
= C
(B)
xe
y
+
y
2
= C
(C)
ye
x
+
x
2
= C
(D)
ye
y
+
x
2
= C
Solution :
The given differential equation is:
Therefore, option (C) is correct.
NCERT Solutions for Class 12 Maths Chapter 9 Miscellaneous Exercise FAQs
What is a Differential Equation?
A differential equation is an equation that involves one or more derivatives of an unknown function. It describes how a function's rate of change relates to the function itself.
Why are differential equations important?
Differential equations are fundamental in modeling various real-world phenomena, such as population growth, motion, heat conduction, and more.
What types of differential equations are covered in Chapter 9?
Chapter 9 primarily focuses on first-order differential equations and their applications. It introduces concepts like separable variables, linear differential equations, and applications of differential equations.
How to solve a separable differential equation?
A separable differential equation can be solved by separating variables and integrating both sides. This involves isolating variables on one side and integrating with respect to each variable.
What are linear differential equations?
An equation containing a variable, its derivative and a few more functions of degree one is called a linear differential equation.
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