JEE Exam  >  JEE Notes  >  Mathematics (Maths) Class 12  >  NCERT Solutions - Exercise Miscellaneous: Differential Equations

NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q1: For each of the differential equations given below, indicate its order and degree (if defined).
(i) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations   
(ii) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
(iii) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: (i) The differential equation is given as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Thus, its order is two. The highest power raised to NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis one. Hence, its degree is one.
(ii) The differential equation is given as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Thus, its order is one. The highest power raised to NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis three. Hence, its degree is three.
(iii) The differential equation is given as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The highest order derivative present in the differential equation isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations. Thus, its order is four.
However, the given differential equation is not a polynomial equation. Hence, its degree is not defined.

Q2: For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
(ii) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
(iii) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
(iv) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: (i) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Again, differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, on substituting the values of NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsand NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
⇒ L.H.S. ≠ R.H.S.
Hence, the given function is not a solution of the corresponding differential equation.
(ii) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Again, differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, on substituting the values of NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsand NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the L.H.S. of the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is a solution of the corresponding differential equation.
(iii) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Again, differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value of NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations in the L.H.S. of the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is a solution of the corresponding differential equation.
(iv) NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating both sides with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value of NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin the L.H.S. of the given differential equation, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given function is a solution of the corresponding differential equation.

Q3: Form the differential equation representing the family of curves given by NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationswhere a is an arbitrary constant.
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
From equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
On substituting this value in equation (3), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the differential equation of the family of curves is given as NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q4: Prove that NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis the general solution of differential equationNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations, where c is a parameter.
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is a homogeneous equation. To simplify it, we need to make the substitution as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of y and NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of I1 and I2 in equation (3), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, equation (2) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given result is proved.

Q5: Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Ans: The equation of a circle in the first quadrant with centre (a, a) and radius (a) which touches the coordinate axes is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating equation (1) with respect to x, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the value of a in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the required differential equation of the family of circles is NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q6: Find the general solution of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q7: Show that the general solution of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis given by (x    1) = (1 – y – 2xy), where is parameter
Ans:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the given result is proved.

Q8: Find the equation of the curve passing through the point NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationswhose differential equation is, NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The differential equation of the given curve is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The curve passes through point NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
On substituting NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the required equation of the curve is NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q9: Find the particular solution of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations, given that y = 1 when x = 0
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting these values in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, y = 1 at x = 0.
Therefore, equation (2) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required particular solution of the given differential equation.

Q10: Solve the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Differentiating it with respect to y, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
From equation (1) and equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q11: Find a particular solution of the differential equationNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations, given that = – 1, when x = 0 (Hint: put xy = t)
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting the values of xand NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, y = –1 at = 0.
Therefore, equation (3) becomes:
log 1 = 0 – 1 C
⇒ C = 1
Substituting C = 1 in equation (3) we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required particular solution of the given differential equation.

Q12: Solve the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This equation is a linear differential equation of the form
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q13: Find a particular solution of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations, given that y = 0 when NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This equation is a linear differential equation of the form
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now,NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, equation (1) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsin equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required particular solution of the given differential equation.

Q14: Find a particular solution of the differential equationNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations, given that y = 0 when x = 0
Ans: 
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting this value in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, at x = 0 and y = 0, equation (2) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting C = 1 in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required particular solution of the given differential equation.

Q15: 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 20000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009
Ans: 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.
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
log = 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 = 25000.
Therefore, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
In the year 2009, t = 10 years.
Now, on substituting the values of t, k, and C in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the population of the village in 2009 will be 31250.

Q16: The general solution of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis
A. xy = C                    
B. = Cy2                        
C. = Cx                   
D. y = Cx2
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Integrating both sides, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the correct answer is C.

Q17: The general solution of a differential equation of the type NCERT Solutions Class 12 Maths Chapter 9 - Differential Equationsis
A. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations   
B. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
C. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations 
D. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The integrating factor of the given differential equationNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the differential equation is given by,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the correct answer is C.

Q18: The general solution of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations is
A. xey + x2 = C
B. xey + y2 = C
C. yex + x2 = C
Dyey + x2 = C
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is a linear differential equation of the form
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the correct answer is C.

The document NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

1. What is a differential equation?
Ans. A differential equation is an equation that relates a function with one or more of its derivatives. It describes how a function changes as a result of some input or conditions.
2. What are the types of differential equations?
Ans. Differential equations can be classified into various types such as ordinary differential equations (ODEs) and partial differential equations (PDEs) based on the number of independent variables involved.
3. How are differential equations solved?
Ans. Differential equations can be solved using various methods such as separation of variables, integrating factors, substitution methods, and series solutions, depending on the type and complexity of the equation.
4. What are the applications of differential equations?
Ans. Differential equations have wide applications in various fields such as physics, engineering, biology, economics, and chemistry to model and analyze various phenomena and processes.
5. How can I practice solving differential equations?
Ans. To practice solving differential equations, you can work on examples from textbooks, online resources, and solve past exam papers to improve your understanding and problem-solving skills in this area.
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