Question

Solve $y(1+xy)dx+x(1-xy)dy=0$.

Answer 1

$(1+xy)ydx+x(1-xy)dy=0$

$ydx+x{y}^{2}dx+xdy-x^{2}ydy=0$

$(xdy+ydx)=-x{y}^{2}dx+x^{2}ydy$

$d\left(xy\right)=xy(xdy-ydx)$

$\frac{d\left(xy\right)}{xy}=(xdy-ydx)$

$-\frac{d\left(xy\right)}{xy}=y^2\left(\frac{ydx-xdy}{y^2}\right)=y^{2}d\left(\frac{x}{y}\right)$

$\int -\frac{d\left(xy\right)}{x^2{y}^2}=\int \frac{y}{x}d\left(\frac{x}{y}\right)$

$-1(-1/xy)=in\left(x/y\right)+c$

$in\left(x/y\right)=c_1+1/xy$

$\frac{x}{y}=e{c}_1+1/xy$