NCERT Solutions - Exercise 9.5: Differential Equations

# NCERT Solutions Class 12 Maths Chapter 9 - Differential Equations

Q1:
Ans: The given differential equation is
This is in the form of

The solution of the given differential equation is given by the relation,

Therefore, equation (1) becomes:

This is the required general solution of the given differential equation.

Q2:
Ans: The given differential equation is

The solution of the given differential equation is given by the relation,

This is the required general solution of the given differential equation.

Q3:
Ans: The given differential equation is:

The solution of the given differential equation is given by the relation,

This is the required general solution of the given differential equation.

Q4:
Ans: The given differential equation is:

The general solution of the given differential equation is given by the relation,

Q5:
Ans: The given differential equation is:

This equation is in the form of:

The general solution of the given differential equation is given by the relation,

Therefore, equation (1) becomes:

Q6:
Ans: The given differential equation is:

This equation is in the form of a linear differential equation as:

The general solution of the given differential equation is given by the relation,

Q7:
Ans: The given differential equation is:

This equation is the form of a linear differential equation as:

The general solution of the given differential equation is given by the relation,

Substituting the value of in equation (1), we get:

This is the required general solution of the given differential equation.

Q8:
Ans:

This equation is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Q9:
Ans:

This equation is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Q10:
Ans:

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Q11:
Ans:

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Q12:
Ans:

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Q13:
Ans: The given differential equation is
This is a linear equation of the form:

The general solution of the given differential equation is given by the relation,

Now,
Therefore,

Substituting C = –2 in equation (1), we get:

Hence, the required solution of the given differential equation is

Q14:
Ans:

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Now, y = 0 at x = 1.
Therefore,

Substituting  in equation (1), we get:

This is the required general solution of the given differential equation.

Q15:
Ans: The given differential equation is
This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Now,
Therefore, we get:

Substituting C = 4 in equation (1), we get:

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 be
According to the given information:

This is a linear differential equation of the form:

The general solution of the given differential equation is given by the relation,

Substituting in equation (1), we get:

The curve passes through the origin.
Therefore, equation (2) becomes:
1 = C
⇒ C = 1
Substituting C = 1 in equation (2), we get:

Hence, the required equation of curve passing through the origin is

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
According to the given information:

This is a linear differential equation of the form:

The general equation of the curve is given by the relation,

Therefore, equation (1) becomes:

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:

This is the required equation of the curve.

Q18: The integrating factor of the differential equation is
A. ex
B. ey
C.
D. x
Ans: The given differential equation is:

This is a linear differential equation of the form:

The integrating factor (I.F) is given by the relation,

Hence, the correct answer is C.

Q19: The integrating factor of the differential equation  is
A.
B.
C.
D.
Ans: The given differential equation is:

This is a linear differential equation of the form:

The integrating factor (I.F) is given by the relation,

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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## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

## 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.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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