Need help with this differential equation using separation of variables.

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Homework Help Overview

The problem involves a differential equation expressed as dy/dx = (xy + 2y - x - 2)/(xy - 3y + x - 3). The original poster seeks assistance in starting the solution process, particularly with the separation of variables technique.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster expresses uncertainty about factoring the numerator and denominator of the equation. Some participants suggest that both can indeed be factored, prompting a reevaluation of the initial assumptions. The discussion includes attempts to separate variables after factoring.

Discussion Status

The conversation has progressed with participants confirming the ability to factor the expressions involved. The original poster has successfully separated the variables and is seeking validation on the resulting equation.

Contextual Notes

There is an indication that the original poster is constrained by the requirement to use separation of variables and is unsure about the initial steps due to perceived difficulties in factoring.

lilmul123
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Homework Statement



The differential equation I have is dy/dx = (xy + 2y - x - 2)/(xy - 3y + x - 3). I need help getting started. Neither the top nor the bottom can be factored, so I don't know what to do next. Can anyone give me a push? All I know is that I need to use separation of variables.
 
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lilmul123 said:
Neither the top nor the bottom can be factored, so I don't know what to do next. Can anyone give me a push? All I know is that I need to use separation of variables.
The top and bottom can both be factored ... try again.
 
lilmul123 said:

Homework Statement



The differential equation I have is dy/dx = (xy + 2y - x - 2)/(xy - 3y + x - 3). I need help getting started. Neither the top nor the bottom can be factored, so I don't know what to do next. Can anyone give me a push? All I know is that I need to use separation of variables.

Hint: Your numerator can be factored:

(xy + 2y - x - 2) = y(x + 2) - (x + 2) = (x + 2)(y - 1).

Similarly in the denominator.
 
Oh, of course. Okay, so now I have (x+2)(y-1) on top, and (x-3)(y+1). I have separated those out. Now I have (y+1)/(y-1) dy = (x+2)/(x-3) dx. Is this the correct equation I need to solve?
 
Looking good so far :smile:
 

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