Suppose you deposit 8500 dollars into a savings account earning 5 percent annual interest compounded continuously. To pay for all your music downloads, each year you withdraw $900 in a continuous way.

Let A(t) represent the amount of money in your savings account t years after your initial deposit.

(A) Write the DE model for the time rate of change of money in the account. Also state the initial condition.

dA/dt=? A(0)=?

(B) Solve the IVP to find the amount of money in the account as a function of time.


(C) When will your money run out?


  • $\begingroup$ You need to put in a small deposit of your work shown here before our interest starts.Continuous compounding involves exponential rise of amount? $\endgroup$
    – Narasimham
    Oct 12, 2015 at 6:05
  • $\begingroup$ i have been working through it for the past hour and cannot get it right. I believe that A(0)=8500 because that is original amount of money. However I am not sure if i should be using that in dA/dt... $\endgroup$
    – Pete
    Oct 12, 2015 at 6:09
  • $\begingroup$ maybe something like 8500e^0.05y-900t $\endgroup$
    – Pete
    Oct 12, 2015 at 6:10

1 Answer 1


As per the notions of differential coefficient of continuous derivatives ( compounding takes place each millisecond !),with annual music download payment $M$ deducted.

$$ \frac{dA}{dt} = r\cdot A- M $$

Integrating $$ \log( r A - M) = r \, t + C $$

At start using boundary condition $$ t= 0 , A =P $$

$$ log \,( r p - M) = C $$

$$ log\, \frac{r \, A - M }{r \,P - M} =r\, t $$

$$ A = ( P- M/r) e ^{r\, t} + M/r $$

$ P = 8500 ; r = 0.05 ; M=900; t = $ no of years.

Bank balance using calculus and influence of each constant can be graphed.

  • $\begingroup$ I used this method but answers are still not coming out correct. $\endgroup$
    – Pete
    Oct 12, 2015 at 19:48
  • $\begingroup$ For dA/dt I am getting (0.05)*(8500e^(0.05t)) according to the above and then for A(t) I get 8500e^(0.05t)-900t. $\endgroup$
    – Pete
    Oct 12, 2015 at 20:01
  • $\begingroup$ Also the money never ran out using this so I could not find answer for t in years $\endgroup$
    – Pete
    Oct 12, 2015 at 20:02
  • $\begingroup$ Not sure what that has to do with the question $\endgroup$
    – Pete
    Oct 12, 2015 at 20:43
  • $\begingroup$ My error, please see correction made for annual payment. $\endgroup$
    – Narasimham
    Oct 12, 2015 at 21:26

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