Given that the differential equation dy/dx = (x+y+2)/(x+y+1), by using a suitable substitution to reduce it to a differential equation with separable variable.
How do I get a suitable substitution for this? Any method can be used?
You should first try using the fact x + y + 2 = x + y + 1 + 1 so you will get 1 + 1/(x+y+1).
I'm not sure what kind of DE this is though. If you need an analytic result or can you use a numerical scheme? The DE should be very close to f(x) = x since (x+y+2) is almost equal to (x+y+1) for large enough |x+y|.