Thinking of a problem which is actually a population riddle.

Question

Most of us if not all have encountered a riddle that if there is a pond there are plants of certain instinct that they double their population each and every day and you are asked that in the 23rd day also the half of the pond is covered plants. how many days will it take or in which day the pond will be totally covered with the plants now I wanted to set up a differential equation so the working process is given below. $ p+ (\frac{dp}{dt}) \times \delta t= 2p$(p=, population as function of time) [ treating $\delta t$ as a day ie $\delta t=1$ so we get $dp/dt=p$ now we know upon solving we get $P=P_0e^t$ and if I see my result the population gets multipled by e each and every day not 2. And if I was to be said how many days will it take to double the population then I'll get ln2 ie 23+ln2 days. Which by all sorts isn't correct. Where am I wrong. Now please don't say 1 day is long interval because I could ask the question in seconds if I wanted. Thanks all

Answer 1

I don't think there is really enough information to solve this problem by differential equations. For example you could imagine that the plant doubles size in the daytime and doesn't change at night.

However if you assume that the rate of growth is proportional to the current size then $$\frac{dP}{dt}=kP$$ and so $$P=Ae^{kt}\ .$$ Since the size doubles every day, $$P(t+1)=2P(t)\quad\Rightarrow\quad Ae^{k(t+1)}=2Ae^{kt}\quad\Rightarrow\quad k=\ln2$$ and so $$P=A(2^t)\ .$$