Finding an Interest Rate

Question

Aspyn wishes to invest an unknown sum in an account where interest is compounded continuously. Assuming Aspyn makes no additional deposits or withdrawals, what annual interest rate will allow her investment to double in exactly five years?

I interpret this as the ODE $P'(t) = rP(t)$ with the initial condition $P(5) = 2P(0)$, where $t$ is the number of years the sum has been in the account, $P(t)$ is the account balance at $t$, and $r$ is a constant interest rate to be computed.

I found the general solution $P(t) = Ce^{rt}$, then tried $2P(0) = Ce^{r5}$, leading to $C = 2e^{-5r}P(0)$ and thus the particular solution $P(t) = 2e^{rt - 5r}P(0)$. I could solve this for $r$, but I'm not sure how to get rid of the $P(0)$ without reintroducing $C$. My overall guess $r = \sqrt[5]{2}$ seems to work, but how can I finish finding the solution?

Answer 1

Once you have $P(t)=Ce^{rt}$, the statement that it doubles in five years means $\frac {P(5)}{P(0)}=2$. The value of $C$ does not matter because it divides out. In fact, $C=P(0)$. It doesn't matter whether $P(0)=1$ or $1000$ or something else. Now you have $$2=e^{5r}\\ \log 2=5r\\r=\frac {\log 2}5$$ which is not the same as $r=\sqrt[5]2$. $\sqrt [5]2$ is the factor the account is multiplied by every year, but is not the nominal interest rate that is compounded continuously.