Question

Integrating factors found by inspection

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

Answer 1

$y(x^2 + y^2 − 1)dx + x(x^2 + y^2 + 1)dy = 0\\ (x^2y + y^3 − y)dx + (x^3 + xy^2 + x)dy = 0\\ x^2ydx + y^3dx − ydx + x^3dy + xy^2dy + xdy = 0\\ x^2(ydx+xdy)+y^2(ydx+xdy)+xdy-ydx=0\\ \left[(ydx+xdy)(x^2+y^2)+xdy-ydx =0\right]\frac{1}{x^2+y^2}\\ ydx+xdy+ \frac{xdy-ydx}{x^2+y^2}=0\\ d(xy)+d(\arctan (\tfrac{y}{x}))=0\\ \int d(xy)+ \int d(\arctan (\tfrac{y}{x}))= \int 0\\ xy+ \arctan \tfrac{y}{x}= c$