I am reading a graduate textbook on D.E.'s (Ordinary Differential Equations with Applications by Chicone) and this is one of the review problems in the first chapter. However D.E.'s. However, it has been 5 years since I took a D.E. class and I do not remember many of the techniques. Could you please provide a hint for solving $$x' = x(1-x)?$$ Thank you very much.
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Hint: write $dx/dt = x(1-x)$ then separate the $t$ and $x$ dependent parts to the 2 sides of the equation. – am301 Jan 04 '22 at 15:25
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Hint: separable equation – Angel Jan 04 '22 at 15:27
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Hint: It is also a Bernoulli equation. Compute the DE for $u=1-x^{-1}$. – Lutz Lehmann Jan 04 '22 at 15:34
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1@Alex Why give the answer? The question has no effort put into it. – Angel Jan 04 '22 at 15:50
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@Angel I'm sorry I deleted my comment. – Jan 04 '22 at 16:33
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1@Moo Oh right you can separate it as $\frac {x'}{x(1-x)} = 1$. Thank you. I did think of doing $\frac {x'}{x} = 1-x$ before but of course that doesn't work so I thought it wasn't separable. – user56202 Jan 04 '22 at 16:47
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@am301Thanks, I got it. – user56202 Jan 04 '22 at 16:48
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@Angel I got it thank you. – user56202 Jan 04 '22 at 16:48
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@LutzLehmann Thanks I see it's separable (easy), but I will read up on this method also. – user56202 Jan 04 '22 at 16:49