The question is :-

$\displaystyle y -xy' = x + yy'$

The question asks to find the integrating factor and to solve it further.

As this is a homogeneous equation I tried the formula of $\displaystyle \frac{1}{Mx + Ny}$ and both partial fraction formulas but the answer which is $\displaystyle \frac{1}{x^2 + y^2}$ wouldn't come. I get $\displaystyle \frac{1}{-x^2-y^2}.$

Secondly even by using the answer of I.F I get stuck at final integration. Please do it for me. I dunno that how does the terms $\displaystyle tan^{-1} (\frac{x}{y})$ and $\displaystyle -ln\sqrt{x^2 + y^2}$ come ?