arenceabigns

2020-12-16

$\left(2y+xy\right)dx+2xdy=0$

coffentw

Expert

Here the Differential equations is given by
$\left(2y+xy\right)dx+2xdy=0.$
This can be written as
$y\left(2+x\right)dx+2xdy=0 ⇒ \frac{2+x}{x}dx+\frac{2}{y}dy=0$
$⇒ \left(\frac{2}{x}+1\right)dx+\frac{2}{y}dy=0$

$⇒ d\left(2\mathrm{ln}x+x\right)+d\left(2\mathrm{ln}y\right)=0$

$⇒ d\left(2\mathrm{ln}x+x+2\mathrm{ln}y\right)=0$

$⇒ \int d\left(2\mathrm{ln}x+x+2\mathrm{ln}y\right)=c$

$⇒ 2\mathrm{ln}x+x+2\mathrm{ln}y=c$

$⇒\mathrm{ln}y=\left(c-2\mathrm{ln}x-x\right)/2$

$⇒ y={e}^{\frac{c-2\mathrm{ln}x-x}{2}}$
where cc is a arbitrary constants.

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