Problem 4 - 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.) $$ y^{\prime}=5(100-y) $$

Step-by-Step Solution

Answer

The differential equation represents limited growth.

Step 1: Understand the Form of the Differential Equation

The given differential equation is \( y' = 5(100-y) \). This equation is in the form of a first-order linear differential equation. We notice that the rate of change \( y' \) is proportional to the difference between a constant and the function \( y \).

Step 2: Analyze the Components

Identify the components in the equation: the constant \( 100 \) and the term \( y \). The expression \( (100-y) \) suggests that the rate of change depends on how much \( y \) is less than \( 100 \). This indicates that as \( y \) approaches \( 100 \), the change \( y' \) approaches zero.

Step 3: Determine Growth Type

The structure \( y' = k(A-y) \) (where \( A \) is a constant) is characteristic of the limited growth model. In this model, the population grows rapidly at first and then slows as it approaches a maximum capacity \( A \), due to limiting factors.

Key Concepts

First-Order Linear Differential Equations

First-order linear differential equations are fundamental in the study of calculus and differential equations. These equations involve derivatives of a function with respect to one variable and are characterized by having the first derivative, denoted as \( y' \), as the highest-order derivative. The general form of a first-order linear differential equation is:\[ y' + P(x)y = Q(x) \]In this form, \( P(x) \) and \( Q(x) \) are functions of \( x \) which can be constants or more complex expressions. The key feature of these equations is the linearity in both the dependent variable \( y \) and its derivative \( y' \). This makes them easier to analyze and solve compared to higher-order or nonlinear equations.

• The equation looks linear because \( y \) and \( y' \) are not raised to any power other than one.
• There are various techniques to solve first-order linear equations, like separation of variables and integrating factors.

Understanding the characteristic linear form helps in identifying these equations quickly, as seen in the provided exercise, where the form \( y' = 5(100-y) \) is recognized as a first-order linear differential equation.

Limited Growth Model

The limited growth model is a concept often introduced in biological or economic contexts to describe how certain factors limit the growth of a system over time. The differential equation \( y' = 5(100-y) \) is a classic example of the limited growth model, represented in the form \( y' = k(A-y) \), where \( k \) is a positive constant representing the growth rate and \( A \) is the carrying capacity or maximum sustainable value.

• In this model, the term \( (A-y) \) shows how the growth rate slows as \( y \) approaches \( A \).
• The growth is rapid initially when \( y \) is much smaller than \( A \) and dissipates as \( y \) nears \( A \), due to the reduction in the difference \( A-y \).

A biological analogy might be a population approaching the carrying capacity of its environment, like a fish population in a pond reaching its ecological limit. Understanding these dynamics is crucial for tackling real-world problems where resources or space limit growth.

Rate of Change Analysis

Rate of change analysis helps us understand how a particular quantity evolves over time, a central task in working with differential equations. The rate of change is depicted by the derivative, \( y' \), which gives us an idea of velocity or speed at which a process happens.
In the differential equation \( y' = 5(100-y) \), the rate of change \( y' \) is proportional to \( (100-y) \). This indicates that the closer the value of \( y \) gets to 100, the smaller the rate of change becomes:

• When \( y \) is small, \( y' \) is large, indicating fast growth.
• When \( y \) nears 100, \( y' \) is small, indicating slow growth.

Rate of change analysis is not just about noting the speed; it’s about understanding underlying relationships that cause these changes. This analysis is valuable for forecasting and making informed decisions in fields ranging from engineering to finance.