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A differential equation is a mathematical equation that involves functions and their derivatives. It describes how a particular quantity changes over time. In this problem, we are concerned with how the amount of money in a savings account changes. The differential equation incorporates both the continuous interest earned and the withdrawals made from the account. This helps describe the overall behavior of the account balance over time.

A good example is given by the differential equation derived in the exercise: \[\begin{equation} \frac{df}{dt} = 0.055 f(t) - 300 . \text(Here, \frac{df}{dt} denotes the rate of change of the account balance over time t.)\end{equation}\]

This formulation ensures that all factors influencing the account are considered, including interest growth and steady withdrawals.

Continuous interest means that the interest on the savings account compounds continuously, leading to exponential growth. Unlike simple or annual compound interest, continuous compounding maximizes the effect of interest over time.

The formula for continuous interest is given by: \[\begin{equation} A = P e^{rt}\end{equation}\]where A is the amount in the account after time t, P is the initial deposit, r is the interest rate, and e represents the base of the natural logarithms.

In the exercise, the continuous interest shows as part of the differential equation. The growth rate of the account balance due to interest is given by: \[\begin{equation} 0.055f(t)\end{equation}\]

This term reflects how the account value, f(t), increases over time just because of the continuous interest compounded at a rate of 5.5%.

Withdrawal rate refers to the rate at which money is taken out of the savings account. It is a steady, continuous inflow, impacting the overall balance negatively. In our case, withdrawals are happening at a constant rate of $300 per year, creating a steady decrease in the account balance.

With a continuous annuity, this withdrawal rate is reflected in the differential equation as a constant term: \[\begin{equation} -300\end{equation}\]

The negative sign indicates that this is an outflow or reduction in the account balance. This makes it clear that while interest is trying to grow the balance, withdrawals continuously pull it down, creating a dynamic balance in the account.

A first-order linear differential equation is a type of differential equation defined by its order and linearity. It involves the first derivative of the function and takes the form: \[\begin{equation} \frac{df}{dt} + P(t)f(t) = Q(t)\end{equation}\]

In our case, the equation: \[\begin{equation} \frac{df}{dt} = 0.055 f(t) - 300\end{equation}\]

is first-order because it involves the first derivative, \[\begin{equation}\frac{df}{dt}\end{equation}\]and it is linear because it can be expressed in the standard linear form.First-order linear differential equations are often solved using an integrating factor, ensuring the solution reflects both the growth and decay components (interest vs. withdrawals) impacting the account balance.Understanding this type of equation is crucial for modeling various continuous processes, like our continuous annuity scenario.