Monday, December 12, 2011

Help please ? (differential equations)?

Suppose a rich uncle has left you “S(0) ” dollars, which you deposit in a bank that pays annual interest at the rate of “ r “ compounded continuously. Show that if you make withdrawals amounting to “ d ” dollars per year ( d %26gt; r S(0) ), then the time required to deplete the money in the bank is


t= 1/r ln[ d/d-rS(0) ] . What happens when the annual withdrawn d , is less or equal to S(0) ??|||There are two difficulties with the statement of this problem





1) It is not stated when during the year the withdrawals are made. If the withdrawals are made earlier, then less interest is earned. Is this a lump sum? It seems that this amount it to be withdrawn continuously, not a practical possibility.





2) The final question should be "d is less than or equal to r*S(0)"





The basic equation for interest compounded continuously is





dM/dt = r M





where M is the amount of money at time t. The solution is





M = S(0) e^(rt)





If we assume that the withdrawals are made continuously, then the equation becomes





dM/dt = rM - d





the general solution of which is





M = C e^(rt) + d/r





Applying the initial condition that M(0) = S(0),





M = (S(0)-d/r) e^(rt) + d/r





The money will be depleted when M = 0, or





0 = (S(0)-d/r) e^(rt) + d/r





e^(rt) = d/r / (d/r - S(0)) = d / (d-r*S(0))





or t = (1/r) ln [d / (d-r*S(0))]





If d %26lt; r*S(0), then the balance grows indefinitely. If d=r*S(0) then the balance stays constant. In either case, the balance is never depleted.

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