Chapter 1.1, Problem 16P

Continuously Compounded Interest. The amount of money $P\left(t\right)$ in an interest bearing account in which the principal is compounded continuously at a rate $r$ per annum and in which money is continuously added or subtracted at a rate of $k$ dollars per annum satisfies the differential equation

$\frac{dp}{dt}=rP+k$. (i)

The case $k<0$ corresponds to paying off a loan, while $k>0$ corresponds to accumulating wealth by the process of regular contributions to an interest bearing savings account.

According to the International Institute of Social History (Amsterdam), the amount of money used to purchase Manhattan Island in $1626$ is valued at $\$1050$ in terms of today’s dollars. If that amount were instead invested in an accountthat pays $4\%$ per annum with continuous compounding, whatwould be the value of the investment in $2020$? Compare with the case that interest is paid at $6\%$ per annum.

Show that the solution to Eq. (i), subject to the initial condition $P\left(0\right)=P_0$, is

$P=\left(P_0+\frac{k}{r}\right)e^{rt}-\frac{k}{r}$. (ii)