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

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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?
 
Thread 'Direction Fields and Isoclines'

Thread 'Direction Fields and Isoclines'

I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
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