#### please solve RD sharma class 12 chapter 21 Differential Equation exercise 21.1 question 4 maths textbook solution

Order=2 , Degree=2 , Non-linear

Hint:

The order is the highest numbered derivative in the equation with no negative or fractional power of the dependent variable and its derivatives, while the degree is the highest power to which a derivative is raised.

Given:

$\sqrt{1+\left [ \frac{dy}{dx} \right ]^{2}}=\left [ c\frac{d^{2}y}{dx^{2}} \right ]^{\frac{1}{3}}$

Solution:

In this question, we will raising both the sides the power 6. So we remove the fractional powers of derivatives of the dependent variable y.

So, the above equation becomes

$\left [1+\left [ \frac{dy}{dx} \right ]^{2} \right ]^{3}=\left [ c\frac{d^{2}y}{dx^{2}} \right ]^{2}$

$1+\left ( \frac{dy}{dx} \right )^{2}+3\left ( \frac{dy}{dx} \right )^{2}+3\left ( \frac{dy}{dx} \right )^{4}=c^{2}\left ( \frac{d^{2}y}{dx} \right )^{2}$

So, in this question, the order of the differential equation is 2 and the degree of the differential equation is 2.

In a differential equation, when the dependent variable and their derivatives are only multiplied by constant or independent variable, then the equation is linear.

So, in this question. The dependent variable is y and the term $\frac{dy}{dx}$ is multiplied by itself and many other are also. So the given equation is non-linear.

Therefore, Order=2 , Degree=2 , Non-linear