NCERT Solution Class 12th Maths Chapter – 9 Differential Equations Exercise 9.1
| Textbook | NCERT |
| class | Class – 12th |
| Subject | Mathematics |
| Chapter | Chapter – 9 |
| Chapter Name | Differential Equations |
| grade | Class 12th Maths solution |
| Medium | English |
| Source | last doubt |
NCERT Solution Class 12th Maths Chapter – 9 Differential Equations Exercise 9.1
?Chapter – 9?
✍ Relations and Functions✍
?Exercise – 9.1?
Determine order and degree(if defined) of differential equations given in Exercises 1 to 10.
1. d4y/ dx4 + sin (y′) = 0
Solution:
d4y/dx4 + sin (y′′′) = 0
⇒ y′′′ + sin (y′′′) = 0
The highest order derivative present in the differential equation is y′′′ . Therefore, its order is four.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
2. y’ + 5y = 0
Solution:
The given differential equation is:
y’ + 5y = 0
The highest order derivative present in the differential equation is y’. Therefore, its order is one.
It is a polynomial equation in y’. The highest power raised to y’ is 1. Hence, its degree is one.
3. (ds/dt) 4 + 3s d2s/dt2 = 0
Solution:
(ds/dt)4 + 3s d2s/dt2 = 0
The highest order derivative present in the given differential equation is (d^2s)/(dt)^2` . Therefore,
It is a polynomial equation in d2s/dt2 and ds/dt . The power raised to (d^2s)/(dt^2) is 1
Hence, its degree is one.
4. d2y/(dx2)2 + cos (dy/dx) = 0
Solution:
d2y(dx2)2 + cos (dy/dx) = 0
The highest order derivative present in the given differential equation is d2y/dx2. Therefore, its order is 2.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
5. d2y/(dx2)2 + cos (dy/dx) = 0
Solution:
d2y(dx2)2 + cos (dy/dx) = 0
The highest order derivative present in the given differential equation is d2y/dx2. Therefore, its order is 2.
The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.
6. ( y′′′)2 + (y″)3 + (y′)4 + y5 = 0
Solution:
( y′′′)2 + (y″)3 + (y′)4 + y5 = 0
The highest order derivative present in the differential equation is y′′′. Therefore, its order is three.
The given differential equation is a polynomial equation in y′′′, y′′ , y′.
The highest power raised to y′′′ is 2. Hence, its degree is 2.
7. y′′′ + 2y″ + y′ = 0
Solution:
y′′′ + 2y″ + y′ = 0
The highest order derivative present in the differential equation is y′′′. Therefore, its order is three.
It is a polynomial equation in y′′′, y′′, y′ . The highest power raised to y′′ is 1. Hence, its degree is 1.
8. y′ + y = ex
Solution:
y′ + y = ex
⇒ y′+ y – ex = 0
The highest order derivative present in the differential equation is y’. Therefore, its order is one.
The given differential equation is a polynomial equation in y’ and the highest power raised to y’ is one. Hence, its degree is one.
9. y″ + (y′)2 + 2y = 0
Solution:
y″ + (y′)2 + 2y = 0
The highest order derivative present in the differential equation is y″. Therefore, its order is two.
The given differential equation is a polynomial equation in y″and y’ and the highest power raised to y″ is one.
Hence, its degree is one.
10. y″ + 2y′ + sin y = 0
Solution:
y″ + 2y′ + sin y = 0
The highest order derivative present in the differential equation is y″. Therefore, its order is two.
This is a polynomial equation in y″ and y’ and the highest power raised to y’ is one. Hence, its degree is one.
11. The degree of the differential equation
(d2y/dx2)3 + (dy/dx)2 + sin (dy/dx) + 1 = 0
(A) 3
(B) 2
(C) 1
(D) not defined
Solution:
(d2y/dx2)3 + (dy/dx)2 + sin (dy/dx) + 1 = 0
The given differential equation is not a polynomial equation in its derivatives. Therefore, its degree is not defined.
Hence, the correct answer is D.
12. The order of the differential equation
2x2d2y/dx2 – 3 dy/dx + y = 0 is
(A) 2
(B) 1
(C) 0
(D) not defined
Solution:
2x2d2y/dx2-3 dy/dx + y = 0
The highest order derivative present in the given differential equation is d2y/dx2. Therefore, its order is two.
Hence, the correct answer is A.