Ncert Solutions For Class 12 Maths Ex 9.1

Ncert Solutions For Class 12 Maths Chapter 9 Ex 9.1

Q.1: Find the degree and order of the differential equation \(\frac{\mathrm{d} ^{4}y}{\mathrm{d} x^{4}}\)+sin (ym) = 0.

 

Solution:

\(\frac{\mathrm{d} ^{4}y}{\mathrm{d} x^{4}}\) + sin (y”’) = 0

y”” + sin (y”’) = 0

y”” is the highest order derivative present in the differential equation.

Therefore, the order is four.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

 

 

Q.2: Find the degree and order of differential equation y’ + 5y = 0

Solution:

Given: y’ + 5y = 0

y’ is the highest order derivative present in the differential equation.

Therefore, the order is one.

The given differential equation is a polynomial equation in y’. The highest degree derivative present in the differential equation is y’.

Therefore, the degree is one.

 

 

Q.3: Find the degree and order of differential equation \((\frac{\mathrm{d} s}{\mathrm{d} t})^{4}+3s\frac{\mathrm{d}^{2} s}{\mathrm{d} t}=0\).

Solution:

\((\frac{\mathrm{d} s}{\mathrm{d} t})^{4}+3s\frac{\mathrm{d}^{2} s}{\mathrm{d} t}=0\)

\(\frac{d^{2}s}{dt^{2}}\) is the highest order derivative present in the differential equation.

Therefore, the order is two.

The given differential equation is a polynomial equation in \(\frac{d^{2}s}{dt^{2}}\) and \(\frac{ds}{dt}\).The power raised to \(\frac{d^{2}s}{dt^{2}}\) is 1.

Hence, its degree is one.

 

 

Q.4: Find the degree and order of differential equation \((\frac{d^{2}y}{dx^{2}})^{2}+cos(\frac{dy}{dx})=0\).

Solution:

\((\frac{d^{2}y}{dx^{2}})^{2}+cos(\frac{dy}{dx})=0\)

\(\\\frac{d^{2}y}{dx^{2}}\) is the highest order derivative present in the differential equation.

Therefore, the order is two.

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Q.5: Find the degree and order of differential equation \(\frac{d^{2}y}{dx^{2}} =\cos 3x+\sin 3x\)

Solution:

\(\frac{d^{2}y}{dx^{2}} =\cos 3x+\sin 3x\\ \\ \Rightarrow \frac{d^{2}y}{dx^{2}}-\cos 3x-\sin 3x=0\)

\(\frac{d^{2}y}{dx^{2}}\) is the highest order derivative present in the differential equation.

Therefore, the order is two.

It is a polynomial equation in \(\frac{d^{2}y}{dx^{2}}\) and the power raised to \(\frac{d^{2}y}{dx^{2}}\) is 1.

Hence, its degree is one.

 

 

Q.6: Find the degree and order of differential equation (y”’)2 + (y”)3 + (y’)4+ y5 = 0

Solution: (y”’)2 + (y”)3 + (y’)4+ y5 = 0

y”’ is the highest order derivative present in the differential equation.

Therefore, the order is three.

It is a polynomial equation in y”’, y” and y’.

The power of y”’ is 2.

Hence, its degree is 2.

 

Q.7: Find the degree and order of differential equation y”’ + 2y” + y’ = 0.

Solution:

Given: y”’ + 2y” + y’ = 0.

y”’ is the highest order derivative present in the differential equation.

Therefore, the order is three.

It is a polynomial equation in y”’, y”, and y’.

The power of y”’ is 1.

Hence, its degree is 1.

 

 

Q.8: Find the degree and order of differential equation y’ + y = ex

Solution: y’ + y = ex

\(\boldsymbol{\Rightarrow }\) y’ + y – ex = 0

y’ is the highest order derivative present in the differential equation.

Therefore, the order is one.

It is a polynomial equation in y’.

The power raised to y”’ is 1.

Hence, its degree is 1.

 

 

Q.9: Find the degree and order of differential equation y” + (y’)2 +2y = 0.

Solution:

y” + (y’)2 +2y = 0

y” is the highest order derivative present in the differential equation.

Therefore, the order is two.

It is a polynomial equation in y” + y’.

The power raised to y” is 1.

Hence, its degree is 1.

 

Q.10: The degree of differential equation \((\frac{d^{2}y}{dx^{2}})+(\frac{dy}{dx})^{2}+\sin (\frac{dy}{dx})+1=0\) is:

(i) 3

(ii) 2

(iii) 1

(iv) not defined

Solution:

\((\frac{d^{2}y}{dx^{2}})+(\frac{dy}{dx})^{2}+\sin (\frac{dy}{dx})+1=0\)

The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined.

Hence, the answer is (iv).

 

Q.11: The degree of differential equation: \(2x^{2}\;\frac{d^{2}y}{dx^{2}}-3\frac{dy}{dx}+y=0\)is:

(i) 2

(ii) 1

(iii) 0

(iv) not defined

Solution:

\(\frac{d^{2}y}{dx^{2}}\) is the highest order derivative present in the differential equation.

Therefore, the order is two.

Hence, the correct answer is (i).