Solution needed for first order ODE $\frac{dy}{dx}$$= \frac{(x+3y-5)}{(x-y-1)}$
The equation is neither homogeneous nor linear. It's not variable separable either. 

Substitute $x = t+2$, $y= u+1$, and it becomes $$u' = \dfrac{t+3u}{t-u}$$
which is homogeneous.
BTW, Maple gives the solution as
$$y \left( x \right) =1-{\frac { \left( x-2 \right)  \left( {\rm W} 
\left(2\,c \left( x-2 \right) \right)+2 \right) }{{\rm W} \left(2\,c
 \left( x-2 \right) \right)}}
$$
where $W$ is the Lambert W function.
