Question

variable separable of xydx - (x2+y2+1)dy=0?

Answer 1

\[xydx - (x^2+y^2+1)dy=0\]
\[xydx = (x^2+y^2+1)dy\]
Now put above equations in dy/dx form

Answer 2

Please, note that the differential form which you can write, from your differential equation, is:
\[dF=xy dx-\left( x ^{2}+y ^{2}+1 \right) dy\]
that differential form is not exact, nevertheless, you can get an exact differential form if you multiply \[dF\] by a function \[1/y ^{3}\]. After integrated the new differential form, you can solve it, in order to get a primitive function, that primitive function, is the solution to your differential equation