
Concept explainers
Continuously Compounded Interest. The amount of money P(t) 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
dpdt=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.
Show that the solution to Eq. (i), subject to the initial condition P(0)=P0, is
P=(P0+kr) er t−kr. (ii)
Use Eq. (ii) in Problem 15 to solve Problems 16 and 17.

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Chapter 1 Solutions
Differential Equations: An Introduction to Modern Methods and Applications
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