By a suitable substitution or differentiation, reduce…

By a suitable substitution or differentiation, reduce $(x^2+y^2+2x-2y)\,dx+2(y-1)\,dy=0$ to linear form and find its general integral.

Step 1: Find the integrating factor

$\dfrac{M_y-N_x}{N}=\dfrac{(2y-2)-0}{2y-2}=1\Rightarrow \mu(x)=e^{\int dx}=e^x$.

Step 2: Multiply and verify exactness

$M_1=e^x(x^2+y^2+2x-2y)$, $N_1=2e^x(y-1)$; now $\partial M_1/\partial y=\partial N_1/\partial x=2e^x(y-1)$.

Step 3: Find the potential function

$F=\displaystyle\int N_1\,dy=e^x(y^2-2y)+g(x)$. Matching $\partial F/\partial x=M_1$ gives $g'(x)=e^x(x^2+2x)$, so $g(x)=e^x x^2$ (integrate by parts twice).