Problem 33 - Advanced Engineering Mathematics

Question

The differential equation \(d P / d t=(k \cos t) P\) where \(k\) is a positive constant, is a model of human population \(P(t)\) of a certain community. Discuss an interpretation for the solution of this equation; in other words, what kind of population do you think the differential equation describes?

Step-by-Step Solution

Answer

The equation models a population with periodic fluctuations, likely influenced by seasonal or cyclical factors.

Step 1: Analyze the differential equation

The given equation is \( \frac{dP}{dt} = (k \cos t) P \). It's a first-order linear differential equation where the rate of change of \( P(t) \) is proportional to \( P(t) \). This implies that the population growth rate depends on time in a cosine-like periodic manner.

Step 2: Solve the differential equation

To solve the differential equation, separate the variables: \( \frac{dP}{P} = k \cos t \, dt \). Integrate both sides to find the solution: \( \ln |P| = k \int \cos t \, dt = k \sin t + C \). Exponentiating, we get the general solution \( P(t) = Ce^{k \sin t} \), where \( C \) is a constant.

Step 3: Interpret the equation

The solution \( P(t) = Ce^{k \sin t} \) suggests that the population changes according to a sinusoidal function. This indicates that the population experiences regular periodic fluctuations, growing during certain periods and decreasing during others. Such a model could represent a population that migrates periodically or is influenced by seasonal factors.

Key Concepts

Population Modeling

Population modeling is a fascinating topic where we use mathematics to study how populations of organisms, including humans, change over time. By using differential equations, we can predict potential increases or decreases in population sizes under various conditions. For example, the differential equation \(\frac{dP}{dt}=(k \cos t) P\) is one such model. It helps us understand how a community’s population size \(P(t)\) might change in response to factors that vary with time. When modeling population, we try to account for:

• Births and deaths, which affect the population directly.
• Immigration and emigration, or people moving in and out of the area.
• Environmental factors that can cause the population to increase or decrease.

Using models like the one in the exercise helps researchers and policymakers make informed decisions, such as planning for resource allocation or conservation efforts.

First-Order Linear Differential Equations

First-order linear differential equations are equations that involve the first derivatives (rates of change) of a function and the function itself. This type of equation has the general form \(\frac{dy}{dx} + P(x)y = Q(x)\). They are used in many fields, including population dynamics. In our example, \(\frac{dP}{dt} = (k \cos t) P\), the equation is already in the form typically seen with population models.

This particular equation indicates that:

• The rate of change of the population \(P(t)\) is proportional to the population itself.
• The proportionality factor is \(k \cos t\), which means this rate of change is affected cyclically, like the cosine function itself.

We solve such equations often by separating variables, which involves manipulating the equation to integrate both sides. This gives insights into how \(P(t)\), the population at time \(t\), evolves. Once we solve these, we can predict dynamic population shifts, taking into account time-dependent factors.

Periodic Functions

Periodic functions are mathematical functions that repeat their values in regular intervals or periods. The most common types of periodic functions are the trigonometric functions such as sine and cosine. In the solution given, \(P(t) = Ce^{k \sin t}\), the sine function reveals a periodic behavior in the population size.

This means that:

• Population increases and decreases cyclically, mirroring a wave-like pattern.
• This wave could represent yearly cycles, such as seasonal migrations or breeding patterns, affecting the number of individuals in a population.

Periodic functions are crucial in modeling real-world phenomena where there is regular repetition, such as tidal movements, daylight hours, or climate patterns that impact the sustainability of populations over time.