$$ \frac{dP}{dt} = 0.2P(1-\frac{P}{1000}) $$ I am asked to find out the common inflection point of each non-equilibrium solution. How to do that? (I know the equilibrium points are $P$ = $0$, and $P$ = $1000$)
How to determine the inflection point of non-equilibrium solution?
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ordinary-differential-equations
2 Answers
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The second derivative is $$ \frac{d^2P}{dt^2}=0.2\left(1-2\frac{P}{1000}\right)\,\frac{dP}{dt} $$ which you can now easily set to zero.
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first derivative ( let $ M=1000)$ $$ P-P^2/M$$ second derivative vanishes at inflection point $$ 1- 2 P/M =0,\, P=M/2 = 500 $$