NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.1 - Differential Equations

Question:2 Determine order and degree (if defined) of differential equation $y^{\prime }+5y=0$

Answer:

Given function is
$y^{\prime }+5y=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{\prime }}$
Therefore, the order of the given differential equation $y^{\prime }+5y=0$ is 1
Now, the given differential equation is a polynomial equation in its derivatives and its highest power raised to y ' is 1
Therefore, it's a degree is 1.

Question:3 Determine order and degree (if defined) of differential equation ${\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)}^4+3s\frac{{\mathrm{d}}^{2}s}{\mathrm{d}{t}^2}=0$

Answer:

Given function is
${\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)}^4+3s\frac{{\mathrm{d}}^{2}s}{\mathrm{d}{t}^2}=0$
We can rewrite it as
$(s^{{}^{\prime }}{)}^4+3s.s^{{}^{″}}=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $s^{{}^{″}}$

Therefore, the order of the given differential equation ${\left(\frac{\mathrm{d}s}{\mathrm{d}t}\right)}^4+3s\frac{{\mathrm{d}}^{2}s}{\mathrm{d}{t}^2}=0$ is 2
Now, the given differential equation is a polynomial equation in its derivatives and power raised to s '' is 1
Therefore, it's a degree is 1

Question:4 Determine order and degree (if defined) of differential equation.

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

Answer:

Given function is
${\left(\frac{d^{2}y}{d{x}^2}\right)}^2+\cos \left(\frac{dy}{dx}\right)=0$
We can rewrite it as
$(y^{{}^{″}}{)}^2+\cos y^{{}^{″}}=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{″}}$

Therefore, the order of the given differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^2+\cos \left(\frac{dy}{dx}\right)=0$ is 2
Now, the given differential equation is not a polynomial equation in its derivatives
Therefore, it's a degree is not defined

Question:5 Determine order and degree (if defined) of differential equation.

$\frac{d^{2}y}{d{x}^2}=\cos 3x+\sin 3x$

Answer:

Given function is
$\frac{d^{2}y}{d{x}^2}=\cos 3x+\sin 3x$
$\Rightarrow \frac{d^{2}y}{d{x}^2}-\cos 3x-\sin 3x=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{″}}\left(\frac{d^{2}y}{d{x}^2}\right)$

Therefore, order of given differential equation $\frac{d^{2}y}{d{x}^2}-\cos 3x-\sin 3x=0$ is 2
Now, the given differential equation is a polynomial equation in it's dervatives $\frac{d^{2}y}{d{x}^2}$ and power raised to $\frac{d^{2}y}{d{x}^2}$ is 1
Therefore, it's degree is 1

Question:6 Determine order and degree (if defined) of differential equation $(y^{‴}{)}^2+(y^{″}{)}^3+(y^{\prime }{)}^4+y^5=0$

Answer:

Given function is
$(y^{‴}{)}^2+(y^{″}{)}^3+(y^{\prime }{)}^4+y^5=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{‴}}$

Therefore, order of given differential equation $(y^{‴}{)}^2+(y^{″}{)}^3+(y^{\prime }{)}^4+y^5=0$ is 3
Now, the given differential equation is a polynomial equation in it's dervatives $y^{{}^{‴}},y^{{}^{″}}and{y}^{{}^{\prime }}$ and power raised to $y^{{}^{‴}}$ is 2
Therefore, it's degree is 2

Question:7 Determine order and degree (if defined) of differential equation

$y^{‴}+2y^{″}+y^{\prime }=0$

Answer:

Given function is
$y^{‴}+2y^{″}+y^{\prime }=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{‴}}$

Therefore, order of given differential equation $y^{‴}+2y^{″}+y^{\prime }=0$ is 3
Now, the given differential equation is a polynomial equation in it's dervatives $y^{{}^{‴}},y^{{}^{″}}and{y}^{{}^{\prime }}$ and power raised to $y^{{}^{‴}}$ is 1
Therefore, it's degree is 1

Question:8 Determine order and degree (if defined) of differential equation

$y^{\prime }+y=e^x$

Answer:

Given function is
$y^{\prime }+y=e^x$
$\Rightarrow$ $y^{{}^{\prime }}+y-e^x=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{\prime }}$

Therefore, order of given differential equation $y^{{}^{\prime }}+y-e^x=0$ is 1
Now, the given differential equation is a polynomial equation in it's dervatives $y^{{}^{\prime }}$ and power raised to $y^{{}^{\prime }}$ is 1
Therefore, it's degree is 1

Question:9 Determine order and degree (if defined) of differential equation

$y^{″}+(y^{\prime }{)}^2+2y=0$

Answer:

Given function is
$y^{″}+(y^{\prime }{)}^2+2y=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{″}}$

Therefore, order of given differential equation $y^{″}+(y^{\prime }{)}^2+2y=0$ is 2
Now, the given differential equation is a polynomial equation in it's dervatives $y^{{}^{″}}and{y}^{{}^{\prime }}$ and power raised to $y^{{}^{″}}$ is 1
Therefore, it's degree is 1

Question:10 Determine order and degree (if defined) of differential equation

$y^{″}+2y^{\prime }+\sin y=0$

Answer:

Given function is
$y^{″}+2y^{\prime }+\sin y=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{″}}$

Therefore, order of given differential equation $y^{″}+2y^{\prime }+\sin y=0$ is 2
Now, the given differential equation is a polynomial equation in it's dervatives $y^{{}^{″}}and{y}^{{}^{\prime }}$ and power raised to $y^{{}^{″}}$ is 1
Therefore, it's degree is 1

Question:11 The degree of the differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0$ is

(A) 3

(B) 2

(C) 1

(D) not defined

Answer:

Given function is
${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0$
We can rewrite it as
$(y^{{}^{″}}{)}^3+(y^{{}^{\prime }}{)}^2+\sin y^{{}^{\prime }}+1=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{″}}$

Therefore, order of given differential equation ${\left(\frac{d^{2}y}{d{x}^2}\right)}^3+{\left(\frac{dy}{dx}\right)}^2+\sin \left(\frac{dy}{dx}\right)+1=0$ is 2
Now, the given differential equation is a not polynomial equation in it's dervatives
Therefore, it's degree is not defined

Therefore, answer is (D)

Question:12 The order of the differential equation $2x^2\frac{d^{2}y}{d{x}^2}-3\frac{dy}{dx}+y=0$ is

(A) 2

(B) 1

(C) 0

(D) Not Defined

Answer:

Given function is
$2x^2\frac{d^{2}y}{d{x}^2}-3\frac{dy}{dx}+y=0$
We can rewrite it as
$2x.y^{{}^{″}}-3y^{{}^{\prime }}+y=0$
Now, it is clear from the above that, the highest order derivative present in differential equation is $y^{{}^{″}}$

Therefore, order of given differential equation $2x^2\frac{d^{2}y}{d{x}^2}-3\frac{dy}{dx}+y=0$ is 2

Therefore, answer is (A)