# 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 $y '+ 5y = 0$

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 $(\dfrac{\mathrm{d^2}y}{\mathrm{d}x^2})^2+\cos(\dfrac{\mathrm{d}y}{\mathrm{d}x})=0$