<s>[INST] solving $x\,dx + xy^2\,dx + y\,dy + yx^2\,dy=0$ I have to solve $x\,dx + xy^2\,dx + y\,dy + yx^2\,dy=0$
Dividing by $dx$ we have
$x + xy^2 + yy' + yy'x^2=0$
From where,
$$\frac{yy'}{1+y^2}+\frac{x}{1+x^2}=\frac{y\,dy}{1+y^2}+\frac{x\,dx}{1+x^2}=\\ =\frac{d(y^2+1)}{1+y^2}+\frac{d(x^2+1)}{1+x^2}= \frac{1}{2}d\ln(1+y^2)+\frac{1}{2} d\ln(1+x^2)=\frac{1}{2}d\ln(1+y^2)(1+x^2)=0$$
Let $c=(1+y^2)(1+x^2)$, so our equation becomes:
$$
d\ln c=0
$$
So what should I do here, should I integrate, or should I divide by $dx$?
If I divide by dx I get the expression $2x+2yy'+2xy^2+2x^2yy'=0$ which has $x$, $y$ and $y'$ and doesn't help me get anywhere.
Thanks in advance.
 [/INST] $$d\ln c=0 \implies \ln(c)=K$$
Another Hint
$$x\,dx + xy^2\,dx + y\,dy + yx^2\,dy=0$$
$$(x + xy^2)dx + (y + yx^2)dy=0$$
It's an exact differential...
$$\frac {\partial P}{\partial y}=\frac {\partial Q}{\partial x} \implies 2xy=2xy$$
$$
\begin{cases}
f(x,y)=\int x+xy^2dx \\
f(x,y)=\int y+yx^2dy 
\end{cases}
$$
Therefore
$$\boxed{x^2+y^2+y^2x^2=K}$$
</s>