MHB How to Solve a Differential Equation Using Separation of Variables?

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The discussion focuses on solving the differential equation dy/dx = (xy + 3x - y - 3)/(xy - 4x + 6y - 24) using the separation of variables method. Participants are asked to factor both the numerator and denominator to facilitate the solution process. The goal is to express the solution in the form ((x + 6)/(y + 3))^7. Clarification on the factoring process and the subsequent steps in separation of variables is sought. The thread emphasizes the importance of proper manipulation of the equation to achieve the desired solution format.
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Can someone please help me solve the following using separation of variables:

dy/dx = (xy + 3x -y-3)/(xy -4x+6y-24)

so that the solution is written in the form: ((x+6)/(y+3))^7 =
 
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What do you get when you factor the numerator and denominator of the right side?
 

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