Solve by separation of variables

[hank]

So, I have this problem here which I'm kind of stuck on...

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

Ok, so the first thing I do is get everything on the right sides...

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

Now, where do I go from here? I know I want to integrate both sides, but then I get something like this

Sxy dy - S 3y dy + Sxdy - 3Sdy = Sxy dx + 2Sydx - xdx - 2dx

which would work, but I'm not sure what to do with the multi-variable integration.

Where do I go after separating stuff to their sides?

 

[arthur ohlsten]

dy/dx=[xy+2y-x-2] / [xy-3y+x-3]

let us first factor the terms
[xy+2y-x-2]= y[x+2]-[x+2]
[x+2][y-1]

xy-3y+x-3 = y[x-3]+[x-3]
[x-3][y+1]

dy/dx={ [x+2][y-1]} / {[x-3][y+1]
[y+1]/[y-1] dy = [x+2] /[x-3] dx improper fractions, divide
{1+2/[y-1] dy = { 1+5/[x-3]} dx integrate
y+2ln[y-1] =x+5ln[x-3] +c answer

please check for errors

 

[hank]

AH!

I see now.
I didn't even think about applying division here.

Thanks tons for clearing that up! Once you pointed that out, it was easy.

Thanks again!

 

[arthur ohlsten]

you are welcome
Of course you saw we first factored the terms.
if you have a improper fraction,[numerator and denominator of the same power], you must divide prior to integrating.

Arthur