Chapter 1.1, Problem 17P

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.

Use Eq. (ii) in Problem $15$ to solve Problems $16$ and $17$.

How long will it take to pay off a student loan of $\$20,000$ if the interest paid on the principal is $5\%$ and the student pays $\$200$ per month. What is the total amount of money repaidby the student?

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)