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The general solution of $x d y-y d x=y d y$ is
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Verified Answer
The correct answer is:
$y=A e^{-x / y}$
$x d y-y d x=y d y$
$\begin{aligned} & \Rightarrow \quad-(y d x-x d y)=y d y \\ & \Rightarrow \quad-\left(\frac{y d x-x d y}{y^2}\right)=\frac{y d y}{y^2}\end{aligned}$
$\Rightarrow \quad-\int d\left(\frac{x}{y}\right)=\int \frac{d y}{y}$
$\Rightarrow \quad-\frac{x}{y}=\ln |y|+C$
$\Rightarrow \quad y=A e^{-x / y}$
$\begin{aligned} & \Rightarrow \quad-(y d x-x d y)=y d y \\ & \Rightarrow \quad-\left(\frac{y d x-x d y}{y^2}\right)=\frac{y d y}{y^2}\end{aligned}$
$\Rightarrow \quad-\int d\left(\frac{x}{y}\right)=\int \frac{d y}{y}$
$\Rightarrow \quad-\frac{x}{y}=\ln |y|+C$
$\Rightarrow \quad y=A e^{-x / y}$
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