Question

(xe^xy+2y)dy+(ye^xy)dx​

Answer 1

Seeing the xy term, I think of solving it using the following rule:

If Mdx+Ndy=0 is of the form f1(xy)ydx+f2(xy)xdy=0 and Mx−Ny≠0 then integrating factor

=1Mx−Ny

However,

When I am trying to take x as common from N, I get a term yx and cannot solve the equation.

OR YOU CAN WRITE

yexydx+(xexy+2y)dy=0

exy(ydx+xdy)=−2ydy

exyd(xy)=−d(y2)

d(exy)=−d(y2)

exy=−y2+C