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Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise 21.9 Question 5  Maths textbook Solution.

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Answer: x\left ( x^{2}-3y^{2} \right )=c

Given: here,\left ( x^{2}-y^{2} \right )dx-2xydy=0

To find: we have to find the solution of given differential equation.

Hint: in homogeneous differential equation

Put y=vx and \frac{dy}{dx}=v+x\frac{dv}{dx}

Solution: we have,

\left(x^{2} y^{2}\right) d x-2 x y d y=0 \\

\Rightarrow \frac{d y}{d x}=\frac{x^{2}-y^{2}}{2 x y} \\

Put y=v x \Rightarrow \frac{d y}{d x}=v+\frac{x d v}{d x} \\

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

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

\Rightarrow x \frac{d v}{d x}=\frac{1-3 v^{2}}{2 v} \\

\Rightarrow x \frac{d v}{d x}=\frac{1-3 v^{2}}{2 v} \\

\Rightarrow \int \frac{2 v}{1-3 v^{2}} d v=\int \frac{d x}{x}                                                    [Intergrating\; both\: side]

\begin{aligned} &\Rightarrow \frac{1}{-3} \int \frac{-6 v}{1-3 v^{2}} d v=\int \frac{d x}{x} \\ &\Rightarrow \int \frac{-6 v}{1-3 v^{2}}=-3 \int \frac{d x}{x} \\ &\Rightarrow \log \left|1-3 v^{2}\right|=-3 \log |x|+\log |c| \\ &\Rightarrow \log \left|1-3 v^{2}\right|=-\log \left|x^{3}\right|+\log |c| \\ &\Rightarrow 1-3 v^{2}=\frac{c}{x^{3}} \\ &\Rightarrow x^{3}\left(1-\frac{3 y^{2}}{x^{2}}\right)=C \\ &\Rightarrow x^{3} \frac{\left(x^{2}-3 y^{2}\right)}{x^{2}}=C \\ &\Rightarrow x\left(x^{2}-3 y^{2}\right)=C \end{aligned}

This is required solution.

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