I'm at my wits end, I don't understand why I'm having so much trouble, I think I know the answer and then it never matches the book.

At the beginning of the Gold Rush, the population of Coyote Gulch, Arizona was 365. From then on, the population would have grown by a factor of e each year, except for the high rate of "accidental" death, amounting to one victim per day among every 100 citizens. By solving an appropriate differential equation, determine, as function of time (a) the actual populations of Coyote Gulch t years from the day the Gold Rush began, and (b) the cumulative number of fatalities.

I got part (a) as follows:

$\displaystyle \frac{dP}{dt} = P - \frac{365}{100}P$

$\displaystyle P = 365e^{-2.65t}$

Where the 365 comes from the initial condition in the problem. This answer agrees with the book.

For part (b), I simply considered (a bit morbidly) the dead people as another population. Let F be the number of dead people at time t. Then

$\displaystyle \frac{dF}{dt} = \frac{365}{100}P$

$\displaystyle \frac{dF}{dt} = \frac{365^2}{100}e^{-2.65t}$

$\displaystyle F = \frac{-365^2}{265}e^{-2.65t}+\frac{265}{365^2}$

Where the fraction on the right comes from the initial condition that there are no fatalities at t=0.

The book, on the other hand, has that the answer is $\displaystyle 365(1-e^{-2.65t})$ fatalities in t years, an answer they obviously got by setting $\displaystyle \frac{dF}{dt}=\left(\frac{365}{100}-1\right)P$. My question is, why? I'm having an interpretation issue here.

In my interpretation, the new population of interest is the dead people. They grow at a rate of 365/100 P per year. They don't "undie", or in any way get removed from the population, so why do you include the growth factor of the original population as your "death" factor here?

I can't seem to edit this post anymore, so I will have to post anaddendum:

My answer of F above has a typo. It should be

$\displaystyle F=\frac{-365^2}{265} e^{-2.65t} + \frac{365^2}{265}$

This doesn't change my question or the fact that it doesn't match the book's answer.