Differential Equations

Class 12 NCERT

NCERT

1   Determine order and degree (if defined) of differential equation$\dfrac{\mathrm{d}^4y}{\mathrm{d}x^4}+\sin(y''')=0$

Solution :

$\dfrac{\mathrm{d}^4y}{\mathrm{d}x^4}+\sin(y''')=0$$\\$ $\implies y''''+\sin(y''')=0$$\\$ 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$(\dfrac{\mathrm{d}s}{\mathrm{d}t})^4+3s\dfrac{\mathrm{d^2}s}{\mathrm{d}t^2}=0$

Solution :

$(\dfrac{\mathrm{d}s}{\mathrm{d}t})^4+3s\dfrac{\mathrm{d^2}s}{\mathrm{d}t^2}=0$$\\$ The highest order derivative present in the given differential equation is$\dfrac{\mathrm{d^2}s}{\mathrm{d}t^2}.$Therefore, its order is two.$\\$ It is a polynomial equation in $\dfrac{\mathrm{d^2}s}{\mathrm{d}t^2} and \dfrac{\mathrm{d}s}{\mathrm{d}t}$The power raised to $ \dfrac{\mathrm{d^2}s}{\mathrm{d}t^2}$ is $1$. Hence , its degree is one.

4   Determine order and degree (if defined) of differential equation $(\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2})^2+\cos(\dfrac{\mathrm{d}y}{\mathrm{d}x})=0$

Solution :

$(\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2})^2+\cos(\dfrac{\mathrm{d}y}{\mathrm{d}x})=0$$\\$ The highest order derivative present in the given differential equation is$\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2}.$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 $(\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2})^2=\cos3x+\sin3x$

Solution :

$(\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2})^2=\cos3x+\sin3x$$\\$ $\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2}=\cos3x+\sin3x$$\\$ The highest order derivative present in the differential equation is $\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2}$ Therefore, its order is two. It is a polynomial equation $\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2}$ in and the power raised to $\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2}$ is $1$.$\\$ Hence, its degree is one.

6   Determine order and degree (if defined) of differential equation$\\$ $( y ''')^ 2 + ( y '') ^3 + ( y ')^ 4 + y^ 5 = 0$

Solution :

$( y ''')^ 2 + ( y '') ^3 + ( y ')^ 4 + y^ 5 = 0$$\\$ The highest order derivative present in the differential equation is y'''. Therefore, its order is three.$\\4 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.

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. 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 $y'+ y = e'$

Solution :

$y' + y = e'\\ \Rightarrow y' + y - e' = 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 + 2 y = 0$

Solution :

$ y ' + ( y ') 2 + 2 y = 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   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. This is a polynomial equation in $y''$ and $y'$ and the highest power raised $y''$ to is one.$\\$ Hence, its degree is one.