Differential Equation problem

Solve the given differential equation by separation of variables.

(dy/dx)= (xy+3x-y-3)/(xy-2x+4y-8)

First, I noticed when i divided both sides by the left hand side and multiplied both sides by dx, nothing cancelled or seemed to work.

I got to thinking.

on the right hand side I preformed long division.

i divided xy+3x-y-3 by xy-2x+4y-8.

I get 1 + (5x-5y+5)/(xy-2x+4y-8)

(dy/dx)= 1 + (5x-5y+5)/(xy-2x+4y-8)

I am stuck here. Any help is welcomed and appreciated.

I think a better approach is beginning instead by factoring the RHS. From there it should be clear how to solve via separation of variables.

MidgetDwarf
I think a better approach is beginning instead by factoring the RHS. From there it should be clear how to solve via separation of variables.

wow, i over thought this problem. thanks a lot.

factoring the left hand side.

(y+3)(x-1)/(y-2)(x+4)

Left hand side= (y-2)/(y+3)dy

right hand side=(x-1)/(x+4)dx

then I integrate. the process is kind of lengthy, requiring trivial integration.

I can't take you enough.

$$e^{x}e^{-y} = e^{x-y}$$
$$e^{x}e^{-y} = e^{x-y}$$