Problem 34 - Calculus Concepts

Question

Write a differential equation expressing the information given and, when possible, find a general solution for the differential equation. The rate of change with respect to time \(t\) of the demand \(D\) for a product is decreasing in proportion to the demand at time \(t\).

Step-by-Step Solution

Answer

The general solution is \( D(t) = C_1 e^{-kt} \).

Step 1: Understanding the Problem

We are given that the rate of change of demand \( D \) over time \( t \) is decreasing proportionally to the demand itself. This suggests an exponential decay model, which means as \( D \) increases, the rate at which it decreases rises linearly.

Step 2: Writing the Differential Equation

The problem translates to a differential equation where \( \frac{dD}{dt} = -kD \), where \( k > 0 \) is the proportionality constant. This equation shows that the change in demand \( \frac{dD}{dt} \) is negatively proportional to the current demand \( D \).

Step 3: Solving the Differential Equation

Separate the variables so that we have \( \frac{1}{D} \, dD = -k \, dt \). Integrate both sides: \( \int \frac{1}{D} \, dD = \int -k \, dt \), resulting in \( \ln|D| = -kt + C \).

Step 4: Finding the General Solution

To find \( D \), we exponentiate both sides: \( |D| = e^{C}e^{-kt} \). Let \( C_1 = e^{C} \, (C_1 > 0) \), resulting in the general solution \( D(t) = C_1 e^{-kt} \).

Step 5: Interpreting the Solution

The general solution \( D(t) = C_1 e^{-kt} \) indicates that the demand decreases exponentially over time, characterized by the decay rate \( k \). This model only fits if the demand is initially positive.

Key Concepts

Exponential Decay

Exponential decay describes a situation where a quantity decreases at a rate proportional to its current value. This concept is often associated with processes like radioactive decay or cooling, but it also applies to economic models, like demand functions. In these models, as the current demand for a product increases, the rate at which the demand decreases also rises.
This results in a characteristic mathematical model that resembles an exponential function. If the demand decreases over time, we can represent it using the equation \[ \frac{dD}{dt} = -kD \] where \( D \) is the demand and \( k \) is a positive constant.
This equation tells us that the rate of change in demand \( \frac{dD}{dt} \) is proportional and opposing to the demand itself. Through integration and algebra, we derive the function \[ D(t) = C_1 e^{-kt} \] which proves an exponential decrease over time.

Rate of Change

The rate of change refers to how much a variable, such as demand, changes over time. It is an essential concept in calculus and differential equations.
In our example, the rate of change of demand \( \frac{dD}{dt} \) reflects how rapidly the demand shifts as time progresses. This rate is negative, signifying a decline.
By setting up a differential equation like \( \frac{dD}{dt} = -kD \), we gain deep insights into how demand behaves over time and how external factors or initial conditions can influence it.

• A greater absolute value of \( k \) implies a faster decay rate.
• Modifications in the initial conditions or assumptions can lead to a different demand trajectory.

This differential equation framework helps predict future trends and strategize economic decisions.

Demand Function

A demand function quantifies the relationship between demand for a product and influencing variables like time and price.
In this problem, our analysis conveys that demand decreases over time and the function is determined primarily by time. The differential equation \( \frac{dD}{dt} = -kD \) allows us to understand this temporal relationship by establishing how demand varies as time unfolds.
The equation posits that the demand function \( D(t) \) evolves exponentially, yielding \[ D(t) = C_1 e^{-kt} \] as the solution. It captures scenarios typical in markets where demand diminishes over time, possibly due to factors like saturation or changing consumer interests.

• It implies each time unit decreases demand by a factor influenced by \( e^{-kt} \).
• Understanding this function can help in planning marketing strategies or regulating inventory.

Using the demand function effectively aids businesses in managing dynamics in consumer demand.

Proportionality Constant

A proportionality constant, denoted by \( k \), determines the strength of the relationship between two quantities. In our differential equation \( \frac{dD}{dt} = -kD \), the constant \( k \) governs how quickly demand decreases over time.
The larger the value of \( k \), the steeper the decline in demand. Conversely, a smaller \( k \) indicates a slower decay. This constant is crucial for modeling exercises in fields involving exponential change, such as economics or biology.

• If \( k \) is known, it helps predict future demand with greater accuracy.
• Adjusting \( k \) can simulate different market conditions for analysis.

In summary, successfully identifying the proportionality constant enhances the power of predictive modeling, guiding effective decision-making.