Question

For the differential equation $(1+x)ydx+(1+y)xdy=0$. Show that the solution of this equation is $\log xy+(x-y)=c.$

Answer 1

$(1+x)ydx+(1+y)xdy=0$

$\Rightarrow \frac{dy}{dx}=\frac{(1+x)y}{(1+y)x}\Rightarrow \frac{1+y}{y}\frac{dy}{dx}=\frac{1+x}{x}$
Integrating both sides
$\int \frac{1+y}{y}\frac{dy}{dx}dx=\int \frac{1+x}{x}dx$

$\Rightarrow \log y+y=\log x+x+c\Rightarrow \log \left(\frac{x}{y}\right)+(x-y)+c$