Chapter 9 of the Class 12 NCERT Mathematics Part II textbook, titled "Differential Equations," introduces students to the concepts and techniques of solving differential equations. This chapter covers various methods for finding solutions to differential equations and understanding their applications. Exercise 9.1 focuses on basic problems related to solving differential equations, helping students practice and apply the methods introduced in the chapter.
NCERT Solutions for Class 12 - Mathematics Part II - Chapter 9 Differential Equations - Exercise 9.1
This section provides detailed solutions for Exercise 9.1 from Chapter 9 of the Class 12 NCERT Mathematics Part II textbook. The exercise includes problems that involve solving first-order differential equations using different methods. The solutions are presented step-by-step to help students understand the techniques for solving these equations and applying them effectively.
Determine the order and degree (if defined) of differential equations given in exercises 1 to 10.
Question 1. \frac{d^4y}{dx^4}+\sin(y''')=0
Solution:
Given:
\frac{d^4y}{dx^4}+\sin(y''')=0 Here the order is 4 and degree is not defined because it is not a polynomial function.
Question 2. y' + 5y = 0
Solution:
Given: y' + 5y = 0
Here, the order is 1 and the degree is also 1.
Question 3. (\frac{ds}{dt})^{4}+3s\frac{d^{2}s}{dt^{2}}=0
Solution:
Given:
(\frac{ds}{dt})^{4}+3s\frac{d^{2}s}{dt^{2}}=0 Here, the order is 2 and the degree is 1.
Question 4. (\frac{d^{2}y}{dx^{2}})=\cos(\frac{dy}{dx})=0
Solution:
Given:
(\frac{d^{2}y}{dx^{2}})=\cos(\frac{dy}{dx})=0 Here, the order is 2 and degree is not defined because it is not a polynomial function.
Question 5. (\frac{d^{2}y}{dx^{2}})=\cos3x+\sin3x
Solution:
Given:
(\frac{d^{2}y}{dx^{2}})=\cos3x+\sin3x Here, the order is 2 and degree is 1.
Question 6. (y''')2 + (y'')3 + y(')4 + y5 = 0
Solution:
Given: (y''')2 + (y'')3 + y(')4 + y5 = 0
The order is 3 and the degree is 2.
Question 7. y'' + 2y'' + y' = 0
Solution:
Given: y''' + 2y'' + y' = 0
The order is 3 and the degree is 1
Question 8. y' + y = ex
Solution:
Given: y' + y = ex
The order is 1 and the degree is 1.
Question 9. y'' + (y')2 + 2y = 0
Solution:
Given: y'' + (y')2 + 2y = 0
The order is 3 the degree is 1
Question 10. y'' + 2y' + sin y = 0
Solution:
Given: y'' + 2y' + sin y = 0
The order is 2 and the degree is one.
Question 11. The degree of the differential equation
(\frac{d^2y}{dx^2})^{3}+(\frac{dy}{dx})^{2}+sin(\frac{dy}{dx})+1=0 is
(A) 3 (B) 2 (C) 1 (D) Not defined
Solution:
Given:
(\frac{d^2y}{dx^2})^{3}+(\frac{dy}{dx})^{2}+sin(\frac{dy}{dx})+1=0 Here, the degree is not defined.
So, the correct option is D.
Question 12. The order of the differential equation
2x^1\frac{d^2y}{dx^2}-3\frac{dy}{dx}+y=0 is
(A) 3 (B) 2 (C) 1 (D) Not defined
Solution:
Given:
2x^1\frac{d^2y}{dx^2}-3\frac{dy}{dx}+y=0 Here, the order is 2.
So, the correct option is A.
Summary
Chapter 9 of the Class 12 NCERT Mathematics Part II textbook, "Differential Equations," covers techniques for solving differential equations. Exercise 9.1 focuses on basic problems involving first-order differential equations, emphasizing methods such as separation of variables and integrating factors. The exercise aims to help students practice and apply these methods to find solutions to differential equations effectively.