Chapter 1.1, Problem 18P

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.

Derive Eq. (ii) in Problem $15$ from the discrete approximation to the change in the principal that occurs during thetime interval $[t,t+\Delta t]$,

$P(t+\Delta t)\cong P(t)+(r\Delta t)P(t)+k\Delta t$,

assuming that $P\left(t\right)$ is continuously differential function on $t\ge 0$.

[Hint: Substitute $P\left(t+\Delta t\right)=P\left(t\right)+P\text{'}\left(t\right)\Delta t+(1/2)P\text{'}\text{'}\left(t\right){\left(\Delta t\right)}^2$, where $t<\hat{t}<t+\Delta t$, simplify, divide by $\Delta t$, and let $\Delta t\to 0$.]

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)