#### Need solution for RD sharma maths class 12 chapter 21 Differential Equation exercise 21.1 question 18

Order=1, 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:
$y=x\frac{dy}{dx}+a\sqrt{1+\left ( \frac{dy}{dx} \right )^{2}}$

Solution:

First of all, we will rearrange the above equation as follows
$y-x\frac{dy}{dx}=a\sqrt{1+\left ( \frac{dy}{dx} \right )^{2}}$

Since the equation has rational powers, we need to remove them.

So squaring both sides, we get
$y^{2}-2xy\frac{dy}{dx}+x^{2}\left ( \frac{dy}{dx} \right )^{2}=a^{2}\left [ 1+\left ( \frac{dy}{dx} \right )^{2} \right ]$

Here in this question, the order of the differential equation is 1 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.

Here the dependent variable is y and the term $\frac{dy}{dx}$ is multiplied by itself. So this equation is non-linear differential equation.

Therefore, Order=1, Degree=2, Non-linear