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Why do we plot $dy/dt$ vs $y$ while analyzing autonomous DE and find critical points? Why can't we just plot Direction Field and use it, as to plot both types of graphs we need computer. Is it so that Direction field can not tell us the exact value of the critical point? (I doubt)

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I am sorry, but I am just starting ODE and cannot understand your comment. –  Vikram Jul 3 '12 at 13:22
Nevermind. I think you are referring to this, in which case the answer is: there is no point in drawing the slope/direction field, because it will just be the slopes associated to a single curve translated up and down over the entire plane. It's redundant to copy this curve an infinite number of times. –  anon Jul 3 '12 at 13:25
@Vikram: What exactly are you looking for, Vikram? As I see, is your main question the last sentence? –  Babak S. Jul 3 '12 at 15:20
@BabakSorouh, I am referring Boyce Diprima book, in it they have drawn Direction field as well as y' vs y graph for the Logistic equation, now from y' vs y graph they find Critical points and also whether the critical points are stable or unstable, but I see that this can be found out by observing the Direction field and there was no need for y' vs y graph. So my question is what addition information we got by y' vs y graph? I hope I am not confusing you :P –  Vikram Jul 3 '12 at 16:15

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