Problem 40 - Advanced Engineering Mathematics

Question

Fluctuating Population The differential equation \(d P / d t=\) \((k \cos t) P\), where \(k\) is a positive constant, is a mathematical model for a population \(P(t)\) that undergoes yearly seasonal fluctuations. Solve the equation subject to \(P(0)=P_{0}\). Use a graphing utility to obtain the graph of the solution for different choices of \(P_{0}\).

Step-by-Step Solution

Answer

The solution is \( P(t) = P_0 e^{k \sin t} \).

Step 1: Understand the Differential Equation

We are given the differential equation \( \frac{dP}{dt} = (k \cos t) P \). This is a first-order linear ordinary differential equation, suggesting that population \( P(t) \) changes over time due to a cosine factor multiplied by \( k \) and the current population \( P \). We need to solve this DE to find \( P(t) \).

Step 2: Separate Variables

To solve the differential equation, we start by separating variables. We arrange terms so that all \( P \) terms are on one side and all \( t \) terms on the other side: \( \frac{1}{P} dP = k \cos t \, dt \).

Step 3: Integrate Both Sides

Integrate both sides of the equation: \( \int \frac{1}{P} \, dP = \int k \cos t \, dt \). The left integral becomes \( \ln |P| \), and the right integrates to \( k \sin t + C_1 \), where \( C_1 \) is the integration constant.

Step 4: Solve for Population Function \( P(t) \)

Exponentiate both sides to solve for \( P \): \( P(t) = C e^{k \sin t} \), where \( C \) is a constant determined by the initial condition \( P(0) = P_0 \).

Step 5: Apply Initial Condition

Apply the initial condition \( P(0) = P_0 \) to determine \( C \). At \( t = 0 \), \( P(0) = Ce^{k \sin 0} = C \). Thus, \( C = P_0 \), and the solution becomes: \( P(t) = P_0 e^{k \sin t} \).

Step 6: Graph the Solution

Use a graphing utility to plot \( P(t) = P_0 e^{k \sin t} \) for different values of \( P_0 \). Try values such as \( P_0 = 1, 10,\) and \( 100 \) while keeping \( k \) constant, to observe how the initial population size impacts the population growth and fluctuations over time.

Key Concepts

Population Modeling

Population modeling involves mathematically representing populations to predict their growth or decline over time.
It can take into account various factors like birth rates, death rates, and environmental influences. In the context of our exercise, we're considering a population that fluctuates seasonally.
The differential equation provided, \( \frac{dP}{dt} = (k \cos t) P \), models how a population \( P(t) \) changes over time due to these seasonal variations. Here, \( k \) is a positive constant that scales the impact of these fluctuations.

• Population changes according to the cosine function, which naturally varies between -1 and 1 over its cycle.
• This implies that the population experiences regular increases and decreases predicting a recurring pattern over time.
• The population will not grow or decline indefinitely but will oscillate around a certain mean.

Understanding this model helps in predicting how populations are impacted by seasonal factors in real-world scenarios like wildlife populations in temperate climates.

First-order Linear Differential Equations

A first-order linear differential equation is a type of differential equation that can be expressed in the form \( \frac{dy}{dt} + p(t)y = q(t) \).
In this scenario, the differential equation \( \frac{dP}{dt} = (k \cos t)P \) is already in a separable form.
This provides a straightforward approach to finding an explicit solution for \( P(t) \).Here are key characteristics:

• Linear nature: The solution involves basic calculus techniques like separation of variables and integration.
• Order of Equation: Being first-order means it involves only the first derivative of the function \( P(t) \).
• Separable: We can easily isolate variables and solve using standard integration techniques.

This type of equation is crucial in various fields, including physics, biology, and economics, where it simplifies modeling changes over time.

Separation of Variables

Separation of variables is a technique used to solve differential equations.
It involves rearranging the equation to isolate all terms involving one variable on one side of the equation and all terms involving the other variable on the opposite side.
For first-order linear differential equations like \( \frac{dP}{dt} = (k \cos t)P \), it allows us to break down a complex relationship into parts that can be more easily managed.The process for applying separation of variables involves:

• Move terms involving \( P \): Place all \( P \) terms on one side to achieve \( \frac{1}{P} dP \).
• Handle \( t \) terms: Rearrange all \( t \) related terms including constants to the other side to get \( k \cos t \, dt \).
• Integrate: Solve the integrals of each side separately, which typically involves basic calculus techniques.

This technique simplifies solving many differential equations, leading to solutions that describe how two variables relate over time. Hence, using separation of variables is essential for predicting evolving systems in science and engineering.