Problem 3 - Applied Calculus

Question

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=0.02 y $$

Step-by-Step Solution

Answer

The differential equation represents unlimited growth.

Step 1: Identify the Structure of the Equation

The given differential equation is \( y' = 0.02y \). This equation is in the form \( y' = ky \), where \( k \) is a constant.

Step 2: Recall the Growth Models

In growth models, the differential equation \( y' = ky \) typically represents an exponential growth model. In this model, the growth rate is proportional to the current size.

Step 3: Determine the Type of Growth

The equation \( y' = 0.02y \) suggests that the change in \( y \) (\( y' \)) is proportional to \( y \). Since there is no limiting factor present, such as a carrying capacity, this represents unlimited growth.

Key Concepts

Unlimited Growth

Unlimited growth occurs when a population or quantity grows at a constant rate per unit time with no restrictions on its expansion. In the equation \( y' = 0.02y \), the growth rate is consistently proportional to itself, which implies that it can keep increasing indefinitely.

This type of growth doesn't consider factors that could slow it down, such as limited resources or environmental constraints. Without these limitations, the factor of 0.02 means that for any amount of \( y \), there is a 2% increase over time.

It's a theoretical model that often does not hold true over long periods in nature or business as resources tend to be finite.

Exponential Growth

Exponential growth is a particular type of unlimited growth where the rate of growth of a quantity is proportional to the current amount of that quantity. This is expressed mathematically as \( y' = ky \). Here, \( k \) is a positive constant, indicating the rate of growth.

For example, in the differential equation \( y' = 0.02y \), we see exponential growth because the rate of change of \( y \) is directly proportional to its value.

• The solution often takes the form \( y = Ce^{kt} \), where \( C \) is the initial value of the quantity and \( e \) is the base of the natural logarithm.
• This model applies to populations, investments, and many natural processes where the growth is unrestricted and sustains a continuous rate relative to its size.

A key feature of this growth is its rapid increase over time, which can lead to very large quantities in short periods.

Growth Models

Growth models are mathematical representations used to describe how a quantity increases over time. Among the most common models are:

• Exponential Growth: This model, as described above, illustrates rapid growth with unlimited potential.
• Logistic Growth: In contrast, logistic models include a carrying capacity, limiting the growth as it approaches a maximum sustainable amount.
• Limited Growth: Similar to logistic growth but often lacks a specific overshooting mechanism and may involve direct constraints.

Understanding these models requires recognizing the structure of the differential equations involved. The equation \( y' = 0.02y \) aligns with exponential growth due to its form \( y' = ky \), suggesting an absence of limiting factors. Consequently, growth models help predict and analyze behavior in various disciplines, including biology, economics, and sociology, where understanding growth dynamics is essential.