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NCERT Solutions

Class 12

Maths

Chapter 9 Differential Equations Exercise 9.2

NCERT Solutions For Class 12 Maths Chapter 9 Differential Equations Exercise 9.2 - 2025-26

Q: 1. Where can I find reliable, step-by-step NCERT Solutions for Class 12 Maths Chapter 9 (Differential Equations)?

You can find detailed and accurate step-by-step NCERT Solutions for Class 12 Maths Chapter 9, prepared by subject matter experts, on Vedantu. These solutions are fully aligned with the latest CBSE 2025-26 syllabus and provide the correct methodology for solving every problem in the textbook exercises.

Q: 2. What is the standard procedure to verify if a given function is a solution to a differential equation in Chapter 9?

To verify if a function is a solution to a differential equation, you must follow the correct CBSE method:Step 1: Differentiate the given function (e.g., y = f(x)) to find the derivatives required in the differential equation (like y', y'').Step 2: Substitute the original function and its derivatives into the Left-Hand Side (LHS) and/or Right-Hand Side (RHS) of the given differential equation.Step 3: Algebraically simplify the expression. If the LHS equals the RHS, the function is a verified solution.

Q: 3. What is the key difference between a 'general solution' and a 'particular solution' for a differential equation?

A general solution represents an entire family of functions that satisfy the differential equation and will always contain arbitrary constants (like 'C'). The number of these constants is equal to the order of the differential equation. In contrast, a particular solution is a specific solution derived from the general one by using given initial conditions to find the exact value of these constants. Therefore, a particular solution has no arbitrary constants.

Q: 4. Why is understanding the 'order' of a differential equation crucial when finding its NCERT solution?

The order of a differential equation, which is determined by the highest derivative present, is fundamentally important because it dictates the number of arbitrary constants that will appear in its general solution. For instance, a first-order equation will have one constant, while a second-order equation will have two. This knowledge is essential for ensuring your final general solution is complete.

Q: 5. How can a student decide which method to use for solving different types of differential equations in Chapter 9?

The choice of method depends entirely on the form of the equation. As per the NCERT syllabus, the primary methods are:Variable Separable: Use this method if you can rearrange the equation to have all 'y' terms with dy on one side and all 'x' terms with dx on the other.Homogeneous Equations: Apply this if the equation can be expressed in the form dy/dx = F(y/x).Linear Differential Equations: This method is for equations in the standard form dy/dx + Py = Q, where P and Q are functions of x.Identifying the correct form is the first step in any problem.

Q: 6. What does the constant of integration 'C' represent when we solve a differential equation?

The constant of integration 'C' is not just a letter to be added at the end of a solution. It represents the vertical shift of the solution curve. Since the derivative of a constant is zero, there are infinitely many curves (a family of curves) that could be the solution. The constant 'C' accounts for this entire family of potential solutions. A specific value for 'C' is found only when a particular condition is given, leading to a particular solution.

Q: 7. Are the NCERT solutions for the Miscellaneous Exercise of Chapter 9 significantly different from those in the main exercises?

Yes, the questions in the Miscellaneous Exercise often require a more comprehensive understanding of the chapter. They might involve combining multiple concepts or using the solution methods in more complex scenarios. While the fundamental principles remain the same, these problems test your ability to apply knowledge from the entire chapter to solve non-standard questions, which is excellent practice for board exams.

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Class 12 Maths Chapter 9 Questions and Answers - Free PDF Download

In NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.2, you’ll learn how to solve different types of differential equations step by step. This part of the chapter helps you understand new ways to work with equations, so you can handle even tricky math problems with confidence.

Table of Content

Struggling with how to find general and particular solutions? Don’t worry—these solutions by Vedantu break down each question simply, with clear explanations and solved examples. You can also check out the latest Class 12 Maths syllabus to see how this topic fits in your exam.

With handy downloadable PDFs, these NCERT Solutions make revision much easier before exams. Mastering this chapter will help you score better and build a strong foundation for competitive exams too. This chapter carries 7 marks in your CBSE exam.

Access NCERT Solutions for Maths Class 12 Chapter 9 - Differential Equations

```html 1. Verify that the given function is a solution of the corresponding differential equation: y = e^x + 1 ; y'' - y' = 0
Summary: Differentiate y = e^x + 1, find y' and y'', and substitute into the equation to check if both sides match.

Given: y = e^x + 1
First derivative: dy/dx = e^x
Second derivative: d/dx (e^x) = e^x
y'' - y' = e^x - e^x = 0
The equation holds, so this function is a solution.

2. Verify that the given function is a solution of the corresponding differential equation: y = x² + 2x + C ; y' - 2x - 2 = 0
Summary: Differentiate y = x² + 2x + C, find y', and check if it satisfies the given equation.

Given: y = x² + 2x + C
First derivative: y' = 2x + 2
Substitute in the equation: (2x + 2) - 2x - 2 = 0
So, y' - 2x - 2 = 0; thus, the function is a solution.

3. Verify that the given function is a solution of the corresponding differential equation: y = cos x + C ; y' + sin x = 0
Summary: Take the derivative of y, substitute into the equation, and check if the equation equals zero.

Given: y = cos x + C
First derivative: y' = -sin x
Substitute: (-sin x) + sin x = 0
The function fits the equation, so it's a solution.

4. Verify that the given function is a solution of the corresponding differential equation: y = √(1 + x²) ; y' = (x y)/(1 + x²)
Summary: Differentiate y = √(1 + x²), express y' in terms of y, and verify the given relation.

Given: y = √(1 + x²)
y' = (1/2√(1 + x²)) × (2x) = x / √(1 + x²)
Multiply and divide by √(1 + x²): y' = x y / (1 + x²)
Substitute and confirm both sides are equal.

5. Verify that the given function is a solution of the corresponding differential equation: y = Ax ; x y' = y (x ≠ 0)
Summary: Find the derivative of y = Ax, multiply both sides accordingly, and check if the equation is satisfied.

Given: y = Ax
First derivative: y' = A
x y' = xA = Ax = y
Both sides are equal, so this function is a solution.

6. Verify that the given function is a solution of the corresponding differential equation: y = x sin x ; x y' = y + x √(x² - y²) (x ≠ 0 and x > y or x < -y)
Summary: Use the product rule to differentiate y, express everything in terms of x and y, and check the relation.

Given: y = x sin x
y' = x cos x + sin x
x y' = x² cos x + x sin x
We know sin x = y/x, so cos x = √(1 - (y/x)²)
x y' = x² √(1 - (y/x)²) + y = x √(x² - y²) + y
The equation is satisfied. Thus, it is a solution.

7. Verify that the given function is a solution of the corresponding differential equation: x y = log y + C ; y' = y² / (1 - x y) (x y ≠ 1)
Summary: Differentiate both sides, rearrange, and verify if y' matches the expected form.

Given: x y = log y + C
Differentiate using product and chain rules:

d/dx (x y) = y + x y'
d/dx (log y) = y'/y

Thus: y + x y' = y'/y
Multiply both sides by y: y² + x y y' = y'
Rearrange: y' - x y y' = y² ⇒ y'(1 - x y) = y²
So, y' = y² / (1 - x y)
The function is a solution.

8. Verify that the given function is a solution of the corresponding differential equation: y - cos y = x ; (y sin y + cos y + x) y' = 1
Summary: Differentiate both sides, find y', and substitute to check if both sides are equal.

Given: y - cos y = x
Differentiate both sides:

d/dx (y) = y'
d/dx (cos y) = -sin y × y'
d/dx (x) = 1

y' + sin y × y' = 1 ⇒ y'(1 + sin y) = 1 ⇒ y' = 1 / (1 + sin y)
Substitute y' back in: (y sin y + cos y + x) × [1 / (1 + sin y)]
The values match, so it is a solution.

9. Verify that the given function is a solution of the corresponding differential equation: x + y = tan^-1y ; y² y' + y² + 1 = 0
Summary: Differentiate both sides and rearrange equations to check if y' matches the required equation.

Given: x + y = tan^-1y
Differentiate both sides:

d/dx (x + y) = 1 + y'
d/dx (tan^-1y) = y' / (1 + y²)

1 + y' = y' / (1 + y²)
Rearrange: y' [1 - 1/(1 + y²)] = -1
Solve: y'² / (1 + y²) = -1 ⇒ y' = -(y² + 1) / y²
The equation matches: y² y' + y² + 1 = 0.

10. Verify that the given function is a solution of the corresponding differential equation: y = √(a² - x²) , x ∈ ( -a,a ) ; x + y (dy/dx) = 0 (y ≠ 0)
Summary: Differentiate y, substitute y and y' into the equation, and check if the sum is zero.

Given: y = √(a² - x²)
y' = (-x) / √(a² - x²)
Left side: x + y y'
Substitute: x + [√(a² - x²)] × [(-x)/√(a² - x²)] = x - x = 0
Equation is satisfied, so this is a solution.

11. The numbers of arbitrary constants in the general solution of a differential equation of fourth order are:
A differential equation's general solution contains as many arbitrary constants as its order. So, a fourth order equation will have 4 arbitrary constants. Thus, the correct answer is 4.

12. The numbers of arbitrary constants in the particular solution of a differential equation of third order are:
A particular solution does not have any arbitrary constants, so the answer is 0.

``` --- ```html

Understanding Differential Equations in Class 12 Maths Ex 9.2

Exercise 9.2 class 12 maths helps you practice verifying solutions of differential equations.
You'll learn to differentiate given functions and check if they satisfy specific equations.
Know the difference between general solutions (with constants) and particular solutions (no constants).
The number of arbitrary constants equals the order of the differential equation in class 12 maths ex 9.2.
This exercise prepares you for questions that may appear in board exams and entrance tests.

```

FAQs on NCERT Solutions For Class 12 Maths Chapter 9 Differential Equations Exercise 9.2 - 2025-26

1. Where can I find reliable, step-by-step NCERT Solutions for Class 12 Maths Chapter 9 (Differential Equations)?

You can find detailed and accurate step-by-step NCERT Solutions for Class 12 Maths Chapter 9, prepared by subject matter experts, on Vedantu. These solutions are fully aligned with the latest CBSE 2025-26 syllabus and provide the correct methodology for solving every problem in the textbook exercises.

2. What is the standard procedure to verify if a given function is a solution to a differential equation in Chapter 9?

To verify if a function is a solution to a differential equation, you must follow the correct CBSE method:

Step 1: Differentiate the given function (e.g., y = f(x)) to find the derivatives required in the differential equation (like y', y'').
Step 2: Substitute the original function and its derivatives into the Left-Hand Side (LHS) and/or Right-Hand Side (RHS) of the given differential equation.
Step 3: Algebraically simplify the expression. If the LHS equals the RHS, the function is a verified solution.

3. What is the key difference between a 'general solution' and a 'particular solution' for a differential equation?

A general solution represents an entire family of functions that satisfy the differential equation and will always contain arbitrary constants (like 'C'). The number of these constants is equal to the order of the differential equation. In contrast, a particular solution is a specific solution derived from the general one by using given initial conditions to find the exact value of these constants. Therefore, a particular solution has no arbitrary constants.

4. Why is understanding the 'order' of a differential equation crucial when finding its NCERT solution?

The order of a differential equation, which is determined by the highest derivative present, is fundamentally important because it dictates the number of arbitrary constants that will appear in its general solution. For instance, a first-order equation will have one constant, while a second-order equation will have two. This knowledge is essential for ensuring your final general solution is complete.

5. How can a student decide which method to use for solving different types of differential equations in Chapter 9?

The choice of method depends entirely on the form of the equation. As per the NCERT syllabus, the primary methods are:

Variable Separable: Use this method if you can rearrange the equation to have all 'y' terms with dy on one side and all 'x' terms with dx on the other.
Homogeneous Equations: Apply this if the equation can be expressed in the form dy/dx = F(y/x).
Linear Differential Equations: This method is for equations in the standard form dy/dx + Py = Q, where P and Q are functions of x.

Identifying the correct form is the first step in any problem.

6. What does the constant of integration 'C' represent when we solve a differential equation?

The constant of integration 'C' is not just a letter to be added at the end of a solution. It represents the vertical shift of the solution curve. Since the derivative of a constant is zero, there are infinitely many curves (a family of curves) that could be the solution. The constant 'C' accounts for this entire family of potential solutions. A specific value for 'C' is found only when a particular condition is given, leading to a particular solution.

7. Are the NCERT solutions for the Miscellaneous Exercise of Chapter 9 significantly different from those in the main exercises?

Yes, the questions in the Miscellaneous Exercise often require a more comprehensive understanding of the chapter. They might involve combining multiple concepts or using the solution methods in more complex scenarios. While the fundamental principles remain the same, these problems test your ability to apply knowledge from the entire chapter to solve non-standard questions, which is excellent practice for board exams.