The general solution of the differential equation y d x-x d y y =… - Vidyadip

Question

The general solution of the differential equation $\frac{y d x-x d y}{y}=0$ is

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Answer

It is given that $\frac{\mathrm{ydx}-\mathrm{xdy}}{\mathrm{y}}=0$
$\Rightarrow \frac{\mathrm{ydx}-\mathrm{xdy}}{\mathrm{xy}}=0$
$\Rightarrow \frac{1}{x} d x-\frac{1}{y} d y=0$
Integrating both sides, we get,
log|x| - log|y| = log k
$\Rightarrow \log \left|\frac{x}{y}\right|=\log k$
$\Rightarrow \frac{x}{y}=k$
$\Rightarrow \mathrm{y}=\frac{1}{\mathrm{k}} \mathrm{x}$
$\Rightarrow$ y = Cx where C = $\frac{1}{k}$

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