Question

Solve the differential equation: $(x{y}^2+x)dx+(y{x}^2+y)dy=0$

Answer 1

$(x{y}^2+x)dx=-(y{x}^2+y)dy$
$x(y^2+1)dx=-y(x^2+1)dy$

$\frac{xdx}{x^2+1}=\frac{-ydy}{y^2+1}$

$\frac{2xdx}{x^2+1}=\frac{-2ydy}{y^2+1}$

$d\left(\ln \right(1+x^2\left)\right)=d(-\ln (1+y^2\left)\right)$

$\int d\left(\ln \right(1+x^2\left)\right)=\int d(-\ln (1+y^2\left)\right)$

$\ln (1+x^2)=-\ln (1+y^2)+c$
Or
$\ln (1+x^2)+\ln (1+y^2)=c$
Or
$(1+x^2)(1+y^2)=e^c$
Or
$x^2{y}^2+x^2+y^2+1=C$