Question

1. Let S(t) be the value of an investment at time t and let r be the annual interest rate, with interest being compounded after every time interval At. Let k be the annual deposit which has an installment made after each time interval At. Then, the value of the investment at time t + At, i.e. S(t + At), is given by: S(t + At) = S(t) + (rAt)S(t) + kAt Amount at the end of time t Interest added over interval At Installment Payment Subtracting S(t) and dividing both sides by At gives the following: S(t + At) - S(t) = rs(t) + k. At The equation for continuous compounding with continuous deposits is obtained by taking the limit as At → 0; in particular, the resulting initial value problem is: alt=rs + k (1) S(0) = So, where So is the initial amount in the account before any deposits are made. (a) The IVP in equation (1) is a first order linear differential equation. Solve for S(t) by using an appropriate integrating factor. (Note that r and k are constant.) (b) If an annual deposit of $15,000 is made in a continuous way into a retirement account with an annual rate of 10% (with interest compounded continuously) for a period of 40 years, how much money would the investor have accumulated? What is the total that the investor paid into the retirement account and how much interest (in dollars) did the investor earn at the end of the 40 years? (You may assume that the account started with an initial balance of $0.)

Answer 1

t any time t SValue of investment annual intrest rat annual deposit. initial amount in the acount S differential eptio given ds dt s Co) rS k So -the eqution Hiret order inear dfferential equation Py = Q t Integrating faar » e. integrating factor . Soluction oDifferent'i nl ca k. e dE + C K. + C S. e S= So C. S. to C ( () is the Soltion D. E. S=

annual deposit. given оо0'51 to per year o. 40Year O rt K So t o.Ix 40 ISooo e о. j 1so,o00 8 189,722.SI 2.S 8,034,72 g039,722. S anount acumulated afEr 40 years is to nvesbr r retiement is amount Poid Total 600,000 nterest Compounded Continuously The to tal interet earned by joves to r S- 6,00,00о 600, 000 8,039,722.S 7,439,722 .5)