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 rateof 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 1The 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 theequation and the constant and all functions of t (including dt) on the other side. Factoring the right hand side and separating the variables, we havedydt1)dydydtx= 0.08= 0.08 dt.X

The differential equation is given as , y'=d+ry where r is the interest rate and d is the deposit…

Answered: An annuity is a fund into which one…

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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5. When interest is compounded, the interest earned is added to the principal amount so that it may also earn interest. For a 1-year period, the principal amount Q is given by n i Q = (₁ + ²+ )" n Q (0) where i is the annual interest rate (given as a decimal) and n is the number of times during the year that the interest is compounded. Modeling with a Differential Equation tights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). sed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. To lure depositors, banks offer to compound interest at different intervals: semi- annually, quarterly, or daily. A certain bank advertises that it compounds interest con- tinuously. If $100 is deposited initially, formulate a mathematical…
Consider the logistic equation (a) Find the solution satisfying y₁ (0) = 8 and y/₂(0) = -5. yi(t) = 8 Y2(t) (b) Find the time t when y(t) = 4. t= (c) When does y2 (t) become infinite? t=
(6) A bacterial culture contains y(0) grown to y(2) = 250[cells]. Construct and solve the differential equation assuming exponential growth. Determine the number of cells y(t) as a function of time. After how many hours does the bacterial culture 50[cells] at t O[hr]. After t = 2[hr], the bacterial culture has contain 2000 cells?
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. Solve the differential equation for the continuous annuity y(t) with deposit rate d = $5000 and continuous interest rate r = 0.02, subject to the initial condition y(0) = 0 (zero initial value). y= ?
A biologist is tracking Lyme Disease cases in a deer population of 8500 in a state forest in Wisconsin. Suppose at time t = 0 the number D(t) of deer with Lyme Disease is 2250 and is increasing at a rate of 110 deer per month. Assume that D'(t) is proportional to the product of the number of deer who have caught Lyme's disease and of those who have not. (a) Write a differential equation to represent this scenario. Use k for your proportionality constant and D(t) as your function (not D). D'(t) kD(t) (8500 – D(t)) ✓ os = می (b) If you did part (a) correctly, the general solution to your differential equation is D(t) = 8500 1+ Ce-8500kt Find C (keep C exact). C = 25 9 من (c) Find k. (Hint: you do NOT need the function in part (b) to find k). Keep k exact. k 1 125000 (d) How long will it take for 50% of the deer population to develop Lyme Disease? Keep k and C exact until your final calculation. Round your final answer to two decimal places. 13.59 ✓ months
(a) Solve the logistic equation y' = 0.27 - 0.005y2 , y(0) = 50. (b) What is the limiting value yL. (c) How long a time will it take to reach 110% yL?

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