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

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


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,


\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}


\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


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