**Unformatted text preview: **.06 P + K .
dt (2.12) b) Find the solution of (2.12) satisfying P(0) = 0 . (The as yet unknown constant K will
appear in your formula)
c) Using the formula in b) find the value of K that will meet your objective as stated in a).
How much of the final $50,000 consists of deposits and how much consists of interest?
24. a) A woman retires and receives a lump sum of $200,000 from her company's retirement
account. Suppose she deposits this in an account paying 7% continuously compounded
annual interest and withdraws $20,000 a year spread out uniformly throughout the year. By
repeating the arguments that led to equation (2.8), show that the principal P(t ) satisfies the
ODE
dP
= .07 P − 20 ,
dt
where we have used units of $1000.
b) Find the solution of (2.13) satisfying the initial condition P(0) = 200 .
29 (2.13) 2 Differential Equations
c) For how many years can the woman continue withdrawing the $20,000 (usually referred to
as an annuity)?
25. a) A man wants to retire at age 65 and be able to draw $10,000 a year from retirement savings
to supplement his Social Security income and employee retirement benefits. At retirement
he anticipates investing his savings in government securities that should return 5% a year
compounded continuously. By repeating the arguments that led to equation (2.8), show
that after retirement the principal P (t ) in his account satisfies the ODE
dP
= .05 P − 10 ,
dt (2.14) where we have used units of $1000.
b) Determine the general solution of (2.14) and find a formula for the solution with, as yet
undetermined, initial value P0 .
c) How large must the man's initial retirement savings be so that he can continue drawing his
$10,000 income (annuity) for 20 years?
dP
= rP + b . Find a formula for the solution of
dt
this equation with initial condition P(0) = P0 . Check that when b = 0 your answer reduces to
(2.9). 26. Exercises 22 - 25 dealt with ODEs of the form 30...

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