91Ó°ÊÓ

In mathematics, the 'rate of change' of a quantity refers to how that quantity changes over time. In the context of differential equations, it's a measurement of how one variable changes with respect to another variable. For example, in our exercise, the differential equation is given by \[ y' = 0.05y - 10,000 \]. Here, \( y' \) represents the rate of change of the balance \( y \) with respect to time \( t \). By analyzing this, we can understand if the balance in a savings account is increasing or decreasing over time. Plugging in the given values, if after 1 year the balance is \( y = 150,000 \), the rate of change is calculated as follows: \[ y' = 0.05 \times 150,000 - 10,000 = 7,500 - 10,000 = -2,500 \].
Since the rate of change \( y' \) is \( -2,500 \), the balance is decreasing at that rate when \( t = 1 \) year.

An 'equilibrium point' in the context of differential equations refers to a stable state where the system doesn't change over time. For the given differential equation \[ y' = 0.05(y - 200,000) \], we find the equilibrium point by setting the rate of change \( y' \) to zero. This gives: \[ 0 = 0.05(y - 200,000) \] Solving for \( y \) we get: \[ y = 200,000 \]

• This means when the balance reaches \( 200,000 \) dollars, it will neither increase nor decrease.
• The balance is stable at this point.

This implies that any monetary value above or below this equilibrium will adjust over time to reach 200,000 dollars.

When we say that something changes at a 'proportional rate,' we mean that the rate of change is directly related to the current value of the variable in question. In our example, the differential equation is written in the form \( y' = k(y - M) \), where \( k \) and \( M \) are constants. Here, \( k = 0.05 \) and \( M = 200,000 \). Hence, \[ y' = 0.05(y - 200,000) \]. This can be interpreted as follows:

• The factor \( 0.05 \) means that the rate of change of the balance is 5% of the difference between the current balance and the equilibrium point (200,000).
• If the balance is above 200,000, the term \( y - 200,000 \) will be positive, making \( y' \) positive, thus increasing the balance.
• If the balance is below 200,000, the term \( y - 200,000 \) will be negative, making \( y' \) negative, thus decreasing the balance.

This relationship helps in understanding the dynamics of how the balance in the account adjusts over time, always moving towards the equilibrium point of 200,000 dollars.