An annuity is a fund into which one makes equal payments at regular intervals. If the fund earns interest at rate r compounded continuously, and deposits are made continuously at the rate of d dollars per year (a continuous annuity), then the value y(t) of the fund after t years satisfies the differential equation y'= d+ry.t (Do you see why?) Solve the differential equation for the continuous annuity y(t) with deposit rate d = $2000 and continuous interest rate r = 0.08, subject to the initial condition y(0) = 0 (zero initial value).
An annuity is a fund into which one makes equal payments at regular intervals. If the fund earns interest at rate r compounded continuously, and deposits are made continuously at the rate of d dollars per year (a continuous annuity), then the value y(t) of the fund after t years satisfies the differential equation y'= d+ry.t (Do you see why?) Solve the differential equation for the continuous annuity y(t) with deposit rate d = $2000 and continuous interest rate r = 0.08, subject to the initial condition y(0) = 0 (zero initial value).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:An annuity is a fund into which one makes equal payments at regular intervals. If the fund earns interest at rate r compounded continuously, and deposits are made continuously at the rate
of d dollars per year (a continuous annuity), then the value y(t) of the fund after t years satisfies the differential equation y' = d+ ry.t (Do you see why?)
Solve the differential equation for the continuous annuity y(t) with deposit rate d = $2000 and continuous interest rate r = 0.08, subject to the initial condition y(0) = 0 (zero initial value).
Step 1
The first step is to rewrite the equation with the differential as = 2000+ 0.08y. Now, separate the variables with all functions of y (including the differential dy) alone on one side of the
equation and the constant and all functions of t (including dt) on the other side. Factoring the right hand side and separating the variables, we have
dy
dt
1)
dy
dy
dt
x
= 0.08
= 0.08 dt.
X
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