hi
The question in the attachment.
OK, this question will destroy my brain.
I want a start!
Because the original DE is homogeneous, use one of the two standard substitutions such as $\displaystyle y=ux,$ with $\displaystyle dy=x\,du+u\,dx.$ Moreover, the following may prove useful:
$\displaystyle M(x,y)=M(x,ux)=x^{n}M(1,u),$ and the same for $\displaystyle N,$ even to the same power of $\displaystyle x,$ because of the homogeneity of the DE. Then multiply through by your integrating factor, and see if you don't see some integrable combinations pop out at you.