#### Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise 21.9 Question 9 Maths textbook Solution.

Answer: $x^{2}\left ( x^{2}-2y^{2} \right )=C.$

Given:$xy\frac{dy}{dx}=x^{2}-y^{2}$

To solve: we have to solve the given differential equation

Hint: In homogeneous differential equation put  $y=vx$ and  $\frac{dy}{dx}=v+x\frac{dv}{dx}$

Solution: we have,

$xy\frac{dy}{dx}=x^{2}-y^{2}$

$\Rightarrow \frac{dy}{dx}=\frac{x^{2}-y^{2}}{xy}$

It is a homogeneous equation.

$y=vx\Rightarrow \frac{dy}{dx}=v+x\frac{dv}{dx}$

So,$v+x\frac{dv}{dx}=\frac{x^{2}-v^{2}x^{2}}{xvx}$

$\Rightarrow x \frac{d v}{d x}=\frac{1-v^{2}}{v}-v \\$

$\Rightarrow x \frac{d v}{d x}=\frac{1-2 v^{2}}{v} \\$

$\Rightarrow \frac{v}{1-2 v^{2}} d r=\frac{d x}{x} \\$

$\Rightarrow \int \frac{-4 v}{1-2 v^{2}} d v=-4 \int \frac{d x}{x}$

$\Rightarrow \log \left|1-2 v^{2}\right|=-4 \log x+\log c \\$

$\Rightarrow \quad 1-2 \frac{y^{2}}{x^{2}}=\frac{C}{x^{4}}[\text { put } v=y / x] \\$

$\qquad \Rightarrow x^{2}\left(x^{2}-2 y^{2}\right)=C$

This is required solution