Find the general integral of the equation x-y-1+(y-x+2)y'=0

Find the general integral of the equation $x-y-1+(y-x+2)y'=0$.

Step 1: Substitute $u=y-x$

Here $y'=\dfrac{y-x+1}{y-x+2}=\dfrac{u+1}{u+2}$. Since $u'=y'-1$,

\[\frac{du}{dx}=\frac{u+1}{u+2}-1=-\frac{1}{u+2}.\]

Step 2: Separate and integrate

\[(u+2)\,du=-dx \quad\Longrightarrow\quad \frac{u^2}{2}+2u=-x+C.\]

Step 3: Back-substitute $u=y-x$

Multiplying by 2 and expanding $(y-x)^2+4(y-x)+2x=K$:

\[\boxed{\,y^2-2xy+x^2+4y-2x=C.}\]