We describe a savings account with continuous withdrawals that eams interest compounded continuously. Let denote the money in the account, years after it was open. Find the differential
equation and initial condition that satisfies.
Initial amount , annual interest rate , withdrawals at the rate of per month
The differential equation is and the initial condition is .

To model this savings account mathematically, we start with the interest earned and the continuous withdrawals. The interest compounding can be expressed using the formula for continuous compounding, which is (in this case, ). The withdrawals amount to dollars per month, which translates to dollars per year. Therefore, the overall change in the account balance over time can be expressed by the differential equation:

To specify the initial condition, we record the initial amount deposited in the account:

Together, these equations give a complete description of the dynamics of the savings account over time.