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

 

Solve the given differential equation by separation of variables.

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

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Solution:

To solve the differential equation `dy/dx = (xy+2y-x-2)/(xy-3y+x-3)` 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-3y+x-3)`

`(xy-3y+x-3)dy = (xy+2y-x-2)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:

`(x - 3y + x - 3)dy = (x(y+1) - y(x+1) - x + 2)dx`

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

Integrating both sides with respect to their respective variables gives:

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

Solving for y gives:

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

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