The general solution of the differential equation $x^2+y^2-2 x y \frac{d y}{d x}=0$ is (where $C$ is a constant of integration.)

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Aicharya · Accepted Answer

$\begin{aligned} & x^2+y^2-2 x y \frac{d y}{d x}=0 \Rightarrow \frac{d y}{d x}=\frac{x^2+y^2}{2 x y} \Rightarrow v+x \cdot \frac{d v}{d x}=\frac{1+v^2}{2 v} \ & 	ext { [Let } y=n x] \ & \Rightarrow \int \frac{2 v d v}{1-v^2}=\int \frac{d x}{x} \ & \Rightarrow-\log \left|1-v^2ight|+\log |c|=\log |x| \ & \Rightarrow \log \left|\frac{c}{1-v^2}ight|=\log |x| \ & \Rightarrow \frac{c}{1-\frac{y^2}{x^2}}=x \ & \Rightarrow c x=x^2-y^2\end{aligned}$

The general solution of the differential equation $x^2+y^2-2 x y \frac{d y}{d x}=0$ is (where $C$ is a…

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