The death rate in a colony of bees is proportional to the population, and without any new births, half of the bees would die in 20 days. However, the actual population of bees doubles in 30 days. Given that the birth rate of the bees is proportional to the population, find the proportionality constant.
I've tried using the equations with dP/dT=P(k1-k2) and such, and I got ln(2)/30, which is incorrect. Can someone please help me? I honestly have no clue where to go next and have been stuck on this problem for several days now.
Start by looking at the case where no bees are born. Then if $P(T)$ is the population after $T$ days, we have
$$\frac{dP}{dT} = -dT$$
where $d$ is the proportion of bees that die per day. This gives us $P(T) = P(0) e^{-dT}$, and knowing that $P(20) = \frac{1}{2}P(0)$ lets us calculate a value for $d$ (you don't even need to know what the starting population is, because the $P(0)$ terms will cancel each other out).
Then we need to introduce the birth rate, which makes the DE
$$\frac{dP}{dT} = (b - d)T$$
where $d$ is the value already calculated, and $b$ is the proportion of bees born per day. In this case, we get $P(T) = P(0)e^{(b - d)T}$, and we are told that $P(30) = 2P(0)$. You can use that to find a value for $b - d$, and hence for $b$.