JEE Exam  >  JEE Notes  >  Mathematics (Maths) Class 12  >  NCERT Solutions - Exercise 9.5: Differential Equations

NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q1: 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 is in the form of NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, equation (1) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required general solution of the given differential equation.

Q2: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required general solution of the given differential equation.

Q3: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required general solution of the given differential equation.

Q4: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: The given differential equation is:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q5: 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 in the form of:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, equation (1) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q6: 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 in the form of a linear differential equation as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q7: 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 the form of a linear differential equation as:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations 
Substituting the value of 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 general solution of the given differential equation.

Q8: 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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q9: 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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q10: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q11: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q12: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q13: 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 is a linear equation of the form:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The general solution of the given differential equation is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now,NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting C = –2 in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the required solution of the given differential equation is NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q14: NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Ans: 
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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now, y = 0 at x = 1.
Therefore,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required general solution of the given differential equation.

Q15: 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 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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Now,NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting C = 4 in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required particular solution of the given differential equation.

Q16: 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.
Ans: Let F (x, y) be the curve passing through the origin.
At point (x, y), the slope of the curve will beNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
According to the given information:
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 the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Substituting in equation (1), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The curve passes through the origin.
Therefore, equation (2) becomes:
1 = C
⇒ C = 1
Substituting C = 1 in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the required equation of curve passing through the origin isNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q17: 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.
Ans: Let F (x, y) be the curve and let (x, y) be a point on the curve. The slope of the tangent to the curve at (x, y) is NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
According to the given information:
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 equation of the curve is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Therefore, equation (1) becomes:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
The curve passes through point (0, 2).
Therefore, equation (2) becomes:
0 2 – 4 = Ce0
⇒ – 2 = C
⇒ C = – 2
Substituting C = –2 in equation (2), we get:
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
This is the required equation of the curve.

Q18: The integrating factor of the differential equationNCERT Solutions Class 12 Maths Chapter 9 - Differential Equations is
A. ex                                   
B. ey                                   
C. NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations                                                   
D. x
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 integrating factor (I.F) is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the correct answer is C.

Q19: The integrating factor of the differential equation NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations is
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 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 integrating factor (I.F) is given by the relation,
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations
Hence, the correct answer is D.

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 one or more functions and their derivatives. It involves the rate of change of a function with respect to one or more independent variables.
2. What are the types of differential equations?
Ans. Differential equations can be classified into various types, including ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve functions of a single variable, while PDEs involve functions of multiple variables.
3. How are differential equations used in real-life applications?
Ans. Differential equations are widely used in various fields such as physics, engineering, biology, economics, and more. They help in modeling and analyzing real-world phenomena involving rates of change.
4. What is the order of a differential equation?
Ans. The order of a differential equation is the highest order derivative present in the equation. For example, a differential equation involving the second derivative of a function is said to be of second order.
5. How can differential equations be solved?
Ans. Differential equations can be solved using various methods such as separation of variables, integrating factors, substitution, and using specific formulas for different types of differential equations. The choice of method depends on the nature of the equation and its variables.
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