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Let's start by understanding the 'rate of change'. The rate of change tells us how fast something is changing over time. In our context, we are looking at the balance of a savings account. It helps us determine if the balance is increasing or decreasing. For example, when we see the differential equation \( y' = 0.04y + 2000 \), the 'y' represents the balance, and the rate of change is the derivative, denoted as 'y prime' or \( y' \).
By plugging in specific values, like \( y = 10000 \), we calculate how quickly the balance changes. As shown in the exercise, if the balance is \(10,000 after a year, the calculation would be:\[ y' = 0.04(10000) + 2000 \ y' = 400 + 2000 = 2400 \]
This means the balance is increasing at a rate of \)2400 per year. So, in the context of this exercise, the rate of change is positive, indicating the balance is growing.

Another key concept is 'proportional growth'. Proportional growth means that the rate at which something changes is directly proportional to the current value. In our differential equation \( y' = 0.04y + 2000 \), the term \( 0.04y \) indicates this proportional relationship.
We can rewrite the differential equation in the form \( y' = k(y + M) \). By factoring out common terms, we get:\[ y' = 0.04(y + 50000) \]
Here, \( k = 0.04 \) and \( M = 50000 \). This reformulation tells us the rate of change of the account balance is proportional to the current balance of \( y \) plus a constant amount (\$ -50000).
This proportionality makes the growth easier to predict and understand since the balance grows exponentially based on the current amount plus the additional constant.

Let's delve into how our savings account balance evolves over time. The importance of a savings account balance lies in understanding how interest rates and additional deposits influence its growth.
In the given differential equation, \( y(t) \) represents the balance at time \( t \). The differential equation \( y' = 0.04y + 2000 \) governs how this balance changes.
The term \( 0.04y \) represents interest growing at 4% of the current balance, meaning your balance grows as a factor of its size. The term \( 2000 \) could represent annual deposits or additional contributions aside from the interest.
In simple terms, every year, your balance increases by 4% of the current balance plus an extra \(2000. So if you start with \)10,000, it's not just the $10,000 growing but also receiving constant contributions, making compound and continuous growth.
This understanding is crucial for managing and predicting your savings efficiently.