Instructions:

Biology. A population of eight beavers has been introduced into a new wetlands area. Biologists estimate that the maximum population the wetlands can sustain is 60 beavers. After 3 years, the population is 15 beavers. If the population follows a Gompertz growth model, how many beavers will be in the wetlands after 10 years?

Gompertz growth model:

$\displaystyle \frac{dy}{dt}=kyln\frac{60}{y}$

Here's what I did.

$\displaystyle dy=kyln\frac{60}{y}(dt)$

$\displaystyle \frac{dy}{yln\frac{60}{y}}=kydt$

$\displaystyle \int \frac{1}{y(ln60-lny)}dy=\int kdt$

$\displaystyle u=ln60-lny$

$\displaystyle du=-\frac{1}{y}dy$

$\displaystyle -du=\frac{1}{y}dy$

$\displaystyle - \int \frac {1}{u}du = \int kdt$

$\displaystyle -ln|u| = kt + C$

$\displaystyle -|ln60-lny| = e^{kt+c}$

$\displaystyle -ln\frac{60}{y} = Ce^{kt}$

$\displaystyle \frac{60}{y}=e^{-Ce^{kt}}$

$\displaystyle 60=e^{-Ce^{kt}}y$

My values for the variables are:

$\displaystyle y=0, t=0

y=15, t=3

y=?, t=10$

However, the solutions manual states the answer for the general soln is

$\displaystyle y=60e^{-Ce^{kt}}$ which is a difference of signs for the constant in the exponential. I'm not sure where I went wrong but I don't want to start plugging in values until I have the right general solution. Help please?