Population of rabbits

Question

The rate of increase of rabbits on Hare island varies directly with the number of rabbits present at any time. If the initial population is $P_0$ and the population doubles in $Td$ days, how long in terms of $Td$, will the population take to triple?

Answer 1

Hint: "Depends directly" would normally imply $P'(t)=P(t)$, but I think they mean "is proportional to the amout of rabbit", which implies $$P'(t)=kP(t)$$, where $P(t)$ is the population after time $t$. Solve this ODE to find $P(t)=c_0e^{kt}$. Now use your initial data $P(t=0)=P_0$ and your datapoint $P(t=t_d)=2P_0$ to determin $c_0$ and $k$.

EDIT: Finding $c_0$ and $k$.

From your initial data, you can directly conclude that $c_0=P_0$. If you use this for the second data point, we get:

$$2P_0=P_0e^{kt_d}$$ $$\ln2=kt_d$$ $$k=\frac{\ln2}{t_d}$$

Hence, $P(t)=P_0e^{\ln2 \frac{t}{t_d}}=P_0e^{\ln(2^{t/t_d})}=P_0e^{t/t_d}$

In order to find the time to triple $t_3$

$$P(t=t_3)=3P_0=P_02^{t_3/t_d}$$

$$t_3=t_d\cdot \frac{\ln(3)}{\ln(2)}$$