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

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Answer: x^{2}y-xy^{2}=C

Given: \left ( y^{2}-2xy \right )dx=\left ( x^{2}-2xy \right )dy

To find: 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

\left ( y^{2}-2xy \right )dx=\left ( x^{2}-2xy \right )dy

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

It is homogeneous differential equation.

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


\begin{aligned} &\Rightarrow v+x \frac{d v}{d x}=\frac{v^{2}-2 v}{1-2 v} \\ &\Rightarrow x \frac{d v}{d x}=\frac{v^{2}-2 v}{1-2 v}-v \\ &\Rightarrow x \frac{d v}{d x}=\frac{v^{2}-2 v-v+2 v^{2}}{1-2 v} \\ &\Rightarrow x \frac{d v}{d x}=\frac{3 v^{2}-3 v}{1-2 v} \end{aligned}

\Rightarrow \frac{1-2 v}{3\left(v^{2}-v\right)} d v=\frac{d v}{x} \\

\Rightarrow \frac{-(2 v-1)}{3\left(v^{2}-v\right)} d v=\frac{d x}{d x} \\

\Rightarrow \int \frac{2 v-1)}{\left.v^{2}-v\right)} d v=-3 \int \frac{d x}{x} \\                [Integrating \; \; on \: \: both\: \: side]

\Rightarrow \log \left|v^{2}-v\right|=-3 \log |x|+\log c . \\

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

\Rightarrow \frac{y^{2}}{x^{2}}-\frac{y}{x}=\frac{c}{x^{3}} \\

\Rightarrow \frac{y^{2}-x y}{x^{2}}=\frac{c}{x^{3}} \\

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

\Rightarrow x y^{2}-x^{2} y=c \\



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