Give the correct answer and solve the given equation: displaystyle{x}{y}{left.{d}{x}right.}-{left({y}+{2}right)}{left.{d}{y}right.}={0}

Give the correct answer and solve the given equation:
$xydx-\left(y+2\right)dy=0$
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Ayesha Gomez

We have to find the solution for the given differential equation
$xydx=\left(y+2\right)dy.$
We can separate the variables.
So,
$xydx=\left(y+2\right)dy$
$⇒\left(\frac{y+2}{y}\right)dy=xdx$
Integrating both sides we get,
$\int \left(\frac{y+2}{y}\right)dy\int xdx$
$⇒\int \left(1+\frac{2}{y}\right)dy=\int xdx$
$⇒dy+\int \left(\frac{2}{y}\right)dy=\int xdx$
$⇒\int dy+2\int \frac{dy}{y}=\int xdx$
$⇒y+2\mathrm{log}\left(y\right)=\frac{{x}^{2}}{2}+C$, where C is a constant of integration.
Hence, the solution of the given differential equation is,
$y+2\mathrm{log}\left(y\right)=\frac{{x}^{2}}{2}+C$, C is a constant

Jeffrey Jordon