We describe a savings account with continuous withdrawals that eams interest compounded continuously, Let denote the money in the account, t 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

To model the savings account with continuous withdrawals and continuously compounded interest, we can set up a differential equation. The amount of money in the account, , increases due to interest and decreases due to withdrawals.

The interest earned continuously is given by , where is the interest rate, and the withdrawals, which occur at a rate of $500 per month, can be expressed as a yearly rate of per year. Therefore, we have:

Given the initial amount of , the initial condition is:

So, the full system is defined by the differential equation:

with the initial condition:

Did you know that continuous compounding can be modeled using the exponential function? When you have an initial investment, say , accumulating interest continuously at a rate , the formula is:

This means that as time progresses, your investment grows exponentially rather than just linearly! It’s like watching a snowball roll down a hill – exciting and fast-growing!

On a practical note, managing withdrawals effectively is crucial. If you find that your withdrawals exceed your interest earnings for extended periods, consider staggering your withdrawals or investing in a higher interest account. This way, you can keep your savings healthy while still accessing your funds when needed. Always keep an eye on your financial goals!