Question

Solve that
$x(1+y^2)dx+y(1+x^2)dy=0$

Answer 1

$x(1+y^2)dx+y(1+x^2)dy=0$

$xdx(1+y^2)=-ydy(1+x^2)$

$\frac{x}{1+x^2}dx=\frac{-y}{1+y^2}dy$

$\frac{2x}{1+x^2}dx=-\frac{2y}{1+y^2}dy$

Integrating,

$\int \frac{2x}{1+x^2}dx=-\int \frac{2y}{1+y^2}dy$

$log(1+x^2)=-log(1+y^2)+logc$

$log\left[\right(1+x^2\left)\right(1+y^2\left)\right]=logc$

$(1+x^2)(1+y^2)=c$.