Ok
x(dy/dx) = y(y + 1)
dy/ { y(y + 1)} = (1/x)dx
integral dy/{y(y + 1)} = ln x + C
rewrite 1 / (y(y+ 1)) as (y + 1 - y) / (y(y + 1)) = (y+1)/(y(y+1)) - y/(y(y+1)) = 1/y - 1/(y+1)
so integral dy/{y(y + 1)} = integral{ dy/y - dy/(y+ 1) } = ln y - ln(y+1) = ln{ y / (y+1)}
ln{y / (y + 1)} = ln x + C
y / (y + 1) = e^{ln x + C} = e^{ln x}e^{C} = Dx, D = constant = e^C
y / (y + 1) = Dx
y = Dx(y + 1) = Dxy + Dx
y - Dxy = Dx
y(1 - Dx) = Dx
y = Dx / (1 - Dx)............[Ans.]
@Rapidfire: Recording a different constant by the same letter is not going to ruin anybody's day, but surely it is sloppy. I do not understand why anybody would encourage anything but diligent bookkeeping. e^C = 1/C = C? It is a little silly to me.
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As a TA, I have seen students who reuse labels like that on their own lead run into some unnecessary confusion which is why I held contest to your recommendation to the asker to be more relaxed about it given what I have seen. I am sure you are aware that plenty of textbooks, professors, journal articles, etc. use distinct letters in these types of procedures, so by your wording we are all ineffective, but I would rather it be that way for the clarity of the student (or asker). To each his own in the end, but I hope you can understand why that small detail is something I think is worth discussing. Thanks for your input.
