Class 12 Maths NCERT Solutions for Chapter 9 Differential Equations Exercise 9.1
Differential Equations Exercise 9.1 Solutions
1. Determine order and degree (If defined) of differential equation 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. Determine order and degree(if defined) of differential equation 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. Determine order and degree (if defined of differential equation (ds/dt)4 + 3s d2s/dt2 = 0
Solution
The highest order derivative present in the given differential equation is d2s/dt2. Therefore, its order is two .
It is a polynomial equation in d2s/dt2 and ds/dt. The power raised to d2s/dt2 is 1.
Hence, its degree is one.
4. Determine order and degree (if defined) of differential equation d2y/(ds2)2 + cos (dy/dx) = 0
Solution
The highest order derivative present in the given differential equation is d2 y/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. Determine order and degree (if defined) of differential equation d2 y/dx2 = cos 3x + sin 3x
Solution
d2y/dx2 = cos 3x + sin 3x
⇒ d2y/dx2 - cos 3x - sin 3x = 0
The highest order derivative present in the differential equation is d2y/dx2 . Therefore, its order is two.
⇒ d2y/dx2 - cos 3x - sin 3x = 0
The highest order derivative present in the differential equation is d2y/dx2 . Therefore, its order is two.
It is a polynomial equation in d2 y/dx2 and the power raised to d2y/dx2 is 1.
Hence, its degree is one.
6. Determine order and degree (if defined) of differential equation
(y''') + (y'')3 + (y')4 + y5 = 0
Solution
(y''') + (y'')3 + (y')4 + y5 = 0
The highest order derivative present in the differential equation is y'''. Therefore, its order is three.
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'' , and y' .
The highest power raised to y''' is 2. Hence, its degree is 2.
The highest power raised to y''' is 2. Hence, its degree is 2.
7. Determine order and degree(if defined) of differential equation 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.
The highest order derivative present in the differential equation is y'''. Therefore, its order is three.
It is a polynomial equation in y''', y'' and y' . The highest power raised to y''' is 1. Hence, its degree is 1.
8. Determine order and degree(if defined) of differential equation 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.
⇒ 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. Determine order and degree (if defined) of differential equation 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 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. Determine order and degree(if defined) of differential equation 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.
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.
Hence, its degree is one.
11. The degree of the differential equation
(d2y/dx2 )3 + (dy/dx)2 + sin(dy/dx) + 1 = 0
(d2y/dx2 )3 + (dy/dx)2 + sin(dy/dx) + 1 = 0
(A) 3
(B) 2
(C) 1
(D) not defined
(B) 2
(C) 1
(D) not defined
Solution
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 2x2(d2y/dx2) + 3(dy/dx) + y = 0 is
(A) 2
(B) 1
(C) 0
(D) not defined
(B) 1
(C) 0
(D) not defined
Solution
The highest order derivative present in the given differential equation is d2 y/dx2 . Therefore, its order is two.
Hence, the correct answer is A.