Solve the given differential equation by separation of variables. `dy/dx = (xy+3x-y-3)/(xy-2x+4y-8)`

Solve the given differential equation by separation of variables. 

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

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Solution:

To solve the differential equation `dy/dx = (xy+3x-y-3)/(xy-2x+4y-8)` by separation of variables, we first need to rewrite it in the form `f(x)dx = g(y)dy`. To do this, we can multiply both sides by `(xy-2x+4y-8)`

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

Now we can separate the variables by moving all the y terms to one side and all the x terms to the other side. This gives us:

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

`(2y - 2x + 4y - 8) dy = (x^2 + 3x - 3x - 3y - 3) dx`

`(6y - 2x) dy = (x^2 - 6y - 3) dx`

Integrating both sides with respect to their respective variables gives:

`∫(6y - 2x) dy = ∫(x^2 - 6y - 3) dx + C`

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Solving for y gives:

`3y^2 - xy = x^2/2 - 3x + C`

So, the general solution is `3y^2 - xy = x^2/2 - 3x + C` where C is an arbitrary constant.