I am trying to solve a differential equation I found online and the problem is as follows
A population of insects in a region will grow at a rate that is proportional to their current population. In the absence of any outside factors the population will triple in two weeks time. On any given day there is a net migration into the area of 15 insects and 16 are eaten by the local bird population and 7 die of natural causes. If there are initially 100 insects in the area will the population survive? If not, when do they die out?
The differential equation of the solution appears to be $dP/dt = (rP+15)-(16+7) = rP-8$
While what I got was close, my equation relied on time for the daily increase/decrease being $dP/dt = rP-8t$.
My logic for this is that these are factors that take place daily, rather than just a set amount. Why is this wrong?