MHB -2.2.35 Show that dy/dx=(x+3y)/(x-y) is homogeneous. and....

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The discussion revolves around proving that the differential equation dy/dx = (x + 3y) / (x - y) is homogeneous. It is established by dividing both the numerator and denominator by x, resulting in dy/dx = (1 + 3(y/x)) / (1 - (y/x)). This transformation demonstrates the homogeneity of the equation, as it can be expressed in terms of y/x. Participants express uncertainty about the reasoning behind its homogeneity and seek further clarification. The conversation also touches on solving the differential equation and mentions the user's lack of experience with pencil sketches.
karush
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$\dfrac{dy}{dx}=\dfrac{x+3y}{x-y}$
ok well following the book example: divide numerator and denominator by x

$\dfrac{dy}{dx}=\dfrac{1+3\dfrac{y}{x}}{1-\dfrac{y}{x}}$

apparently, thus this is homogeneous but not sure why?

next solve the DE:unsure:
 
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never did any pencil sketch before
so this is my first one
 

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