Problem 17 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose a population is changing according to the equation \(\frac{d P}{d t}=k P-E\), where \(E\) is the rate at which people are emigrating from the country. As established in part (d) of the previous problem, \(P(t)=P_{0} e^{k t}-E t\) is not a solution to this differential equation. (a) Use substitution to solve \(\frac{d P}{d t}=k P-E\). (Your answer ought to agree with that given in part (b).) (b) Verify that \(P(t)=C e^{k t}+\frac{E}{k}\), where \(C\) is a constant, is a solution to the differential equation \(\frac{d P}{d t}=k P-E\)

Step-by-Step Solution

Answer

The solution to the differential equation \(\frac{d P}{d t}=k P-E\) is \(P(t)=\frac{-E}{k}+C e^{kt}\)

Step 1: Solve the differential equation

To solve the first order linear differential equation \(\frac{d P}{d t}=k P-E\), one can use the integrating factor method. First, rewrite the equation to standard form by rearranging: \(\frac{d P}{d t}-k P=-E\). The integrating factor is \(e^{-kt}\). Multiplying the equation with the integrating factor yields: \(e^{-kt}\frac{d P(t)}{d t}-k e^{-kt} P(t)=-E e^{-kt}\). Now, notice that the left-hand side of the equation is the derivative of \(e^{-kt}P(t)\) with respect to \(t\). Using the product rule, integrate both sides with respect to \(t\), resulting in: \(e^{-kt}P(t) = \frac{-E}{k}e^{-kt}+C\). Express \(P(t)\) by multiplying both sides with \(e^{kt}\) to obtain: \(P(t)=\frac{-E}{k}+C e^{kt}\)

Step 2: Substitute the solution

Substitute \(P(t)=\frac{-E}{k}+C e^{kt}\) in \(\frac{d P}{d t}=k P-E\). The derivative of \(P(t)\) is \(\frac{d P(t)}{d t}=kC e^{kt}\). Therefore, \(\frac{d P}{d t}=k(\frac{-E}{k}+C e^{kt})-E=kC e^{kt}-E\), which is \(-E\), same as the original equation. Thus, \(P(t)=\frac{-E}{k}+C e^{kt}\) is indeed a solution of \(\frac{d P}{d t}=k P-E\)

Key Concepts

Integrating Factor Method

The integrating factor method is used for solving linear first-order differential equations. These equations often have the form \(\frac{dy}{dx} + P(x)y = Q(x)\). In our case, to solve \(\frac{dP}{dt} = kP - E\), we rewrite it as \(\frac{dP}{dt} - kP = -E\). The primary step is to find an integrating factor, which is a function we multiply the entire equation by, making the left-hand side a product of a derivative. It essentially helps in easing the integration process.

For our equation, the integrating factor is \(e^{-kt}\). We multiply the whole differential equation by this factor, turning the left side into the derivative of \(e^{-kt}P(t)\). This neat transformation allows us to integrate both sides easily. The integration on the left side simplifies due to the product rule of derivatives:

• The derivative of \(e^{-kt}P(t)\) is \(e^{-kt}\frac{dP}{dt} - ke^{-kt}P\).
• On the right side, we have the integral of \(-Ee^{-kt}\).
• Integrating gives us the solution: \(e^{-kt}P(t) = \int -Ee^{-kt} dt + C\).

To get \(P(t)\) explicitly, multiply through by \(e^{kt}\) resulting in \(P(t) = -\frac{E}{k} + Ce^{kt}\). This method elegantly handled the equation involving both exponential growth and constant emigration.

Exponential Functions

Exponential functions are a fundamental concept in mathematics and arise in various natural phenomena. They are expressed with the equation \(f(x) = ae^{bx}\), where \(a\) and \(b\) are constants, and \(e\) is the base of natural logarithms, approximately 2.718.
In our differential equation solution, the term \(Ce^{kt}\) is an exponential function. It signifies exponential growth or decay, governed by the constant \(k\). This component reveals how a population might change exponentially if the rate \(k\) is positive, representing growth, or negative, indicating decay.

• The constant \(C\) adjusts the initial conditions, providing flexibility for various scenarios.
• If \(C\) is positive and \(k > 0\), the exponential function illustrates population growth over time.
• Conversely, if \(k < 0\), it describes a decreasing scenario.
• The sign and value of \(k\) critically affect how rapidly changes occur in exponential terms.

Understanding \(e^{kt}\) in our context is crucial as it encapsulates the natural growth behavior within the context of our population model.

Population Models

Population models are mathematical representations that describe how populations change over time. Such models include various factors like birth rates, death rates, or in our case, emigration. The differential equation \(\frac{dP}{dt} = kP - E\) bridges these real-world changes with mathematical precision.

This model addresses exponential growth, represented by \(kP\), where populations increase continuously based on a proportionality constant \(k\). It mimics natural growth seen in biological populations under optimal conditions. Meanwhile, \(E\), the emigration term, accounts for individuals leaving the population:

• This introduces a realistic constraint or reduction in population size.
• Such terms help reflect environmental or economic influences.
• The equation balances growth and emigration, leading to the derived solution \(P(t) = Ce^{kt} - \frac{E}{k}\).
• A steady-state or equilibrium can occur if emigration balances the growth, making \(\frac{dP}{dt} = 0\) ultimately.

Population models like this are pivotal, offering insights into population dynamics and helping foresee future scenarios under varying conditions. These models support decision-making in ecology, sociology, and urban planning, where managing population sizes is crucial.