dy/dx=x-y+3/2x-2y+5 is ambiguous. Let's assume dy/dx=(x-y+3)/(2x-2y+5).
Let v=x-y+3, then dv/dx=1-dy/dx, so dy/dx=1-dv/dx, and x-y=v-3, so 2(x-y)+5=2v-6+5=2v-1.
1-dv/dx=v/(2v-1), 2v-1-(2v-1)dv/dx=v, v-1=(2v-1)dv/dx, (2v-1)dv/(v-1)=dx. (2+1/(v-1))dv=dx.
Now we can integrate: 2v+ln(v-1)=x+C, where C is an integration constant.
So 2(x-y+3)+ln(x-y+2)=x+C, x-2y+6+ln(x-y+2)=C.
We can absorb 6 into C: x-2y+ln(x-y+2)=C.