Solve the differential equation: (dy)/(dx) = (x + y + 2)/(2(x + y) - 1)

Put $x+y=v$, so $\frac{dy}{dx} = \frac{dv}{dx}-1$. $\frac{dv}{dx} - 1 = \frac{v+2}{2v-1}$, so $\frac{dv}{dx} = \frac{3v+1}{2v-1}$. Separating variables: $\int\frac{2v-1}{3v+1}\,dv = \int dx$ $\frac{2v-1}{3v+1} = \frac23 - \frac{5}{3(3v+1)}$, so $\frac{2v}{3} - \frac{5}{9}\log|3v+1| = x + c$. Multiplying by 9: $6v - 5\log|3v+1| = 9x + 9c$. Putting $v = x+y$: $6y - 3x - 5\log|3x+3y+1| = C$. Hence the solution is $6y - 3x - 5\log|3x+3y+1| = C$.