Problem 9 - Biocalculus Calculus for the Life Sciences

Question

9\. Suppose a glass of cold water is sitting in a warm room and you place a coin at room temperature \(R\) into the glass. The coin gradually cools down while, at the same time, the glass of water warms up. Newton's law of cooling suggests the following system of differential equations to describe the process $$\frac{d w}{d t}=-k_{m}(w-R) \quad \frac{d p}{d t}=-k_{p}(p-w)$$ where \(w\) and \(p\) are the temperatures of the water and coil (in "C), respectively, and the \(k\) 's are positive constants. $$\begin{array}{l}{\text { (a) Explain the form of the system of differential equations }} \\ {\text { and the assumptions that underlie them. }} \\\ {\text { (b) Use a change of variables to obtain a homogeneous }} \\ {\text { system. }} \\ {\text { (c) What is the general solution to the system you found in }} \\ {\text { part (b)? }} \\ {\text { (d) What is the solution to the original initial-value problem }} \\ {\text { if } w(0)=w_{0} \text { and } p(0)=p_{0} ?}\end{array}$$

Step-by-Step Solution

Answer

Apply Newton's Law and solve the transformed homogeneous system using initial conditions.

Step 1: Understand the Equations

The system of differential equations given is: 1. \( \frac{dw}{dt} = -k_m(w - R) \)2. \( \frac{dp}{dt} = -k_p(p - w) \)These represent Newton's Law of Cooling. The first equation describes how the water's temperature \(w\) changes over time relative to the room temperature \(R\), with a rate proportional to \(w - R\). The second equation indicates how the coin's temperature \(p\) changes relative to the temperature of the water \(w\). The constants \(k_m\) and \(k_p\) represent the cooling rates for water and the coin, respectively.

Step 2: Define Change of Variables

To solve a homogeneous system, introduce new variables: let \( x = w - R \) and \( y = p - R \). This will transform the equations to a new coordinate system centered at the room temperature.

Step 3: Substitute Variables

Substitute \( x \) and \( y \) into the original equations:1. \( \frac{dx}{dt} = -k_m x \) (since \( x = w - R \))2. \( \frac{dy}{dt} = -k_p (y - x) \) (since \( y = p - R \))This transformation results in a homogeneous system of equations where the reference temperature \(R\) has been removed.

Step 4: Solve the Homogeneous System

The transformed differential equations are now:1. \( \frac{dx}{dt} = -k_m x \)2. \( \frac{dy}{dt} = -k_p (y - x) \)The solution to the first equation is \( x(t) = x_0 e^{-k_m t} \). Substitute \( x(t) \) into the second equation to solve for \( y \).

Step 5: Solve Using Initial Conditions

From Step 4, solve the equation \( \frac{dy}{dt} = -k_p(y - x_0 e^{-k_m t}) \). Using methods of integration, we solve for \( y(t) \). We use the initial conditions \( w(0) = w_0 \) and \( p(0) = p_0 \) to ensure that \( x_0 = w_0 - R \) and \( y_0 = p_0 - R \).

Step 6: Find the General Solution

The general solutions are:1. \( w(t) = R + (w_0 - R)e^{-k_m t} \)2. \( p(t) = R + (p_0 - R - \frac{k_p}{k_p - k_m}(w_0 - R) + \frac{k_p}{k_p - k_m}(w_0 - R) e^{-k_m t}) e^{-k_p t} \)

Step 7: Conclusion with Initial Conditions

Substitute the initial conditions into the solutions found in Step 6. The complete solution for the original initial-value problem using the given starting conditions for water and coin temperatures is derived in terms of \( w_0 \), \( p_0 \), and \( R \).

Key Concepts

Differential Equations

Differential equations are powerful mathematical tools used to describe how things change over time. In this exercise, we explore a system of differential equations that captures the dynamics of temperature changes in a glass of water and a coin. The equations are as follows:

• \( \frac{dw}{dt} = -k_m(w - R) \)
• \( \frac{dp}{dt} = -k_p(p - w) \)

These equations are rooted in Newton's Law of Cooling, where the rate of change of temperature depends on the difference from a reference temperature. For the water, this reference is the room temperature \( R \), while for the coin, it's the water's temperature \( w \). The coefficients \( k_m \) and \( k_p \) determine how quickly each body cools down. Understanding these equations is crucial because they predict how the temperatures will evolve over time, a concept essential for solving many practical problems.

Homogeneous System

A homogeneous system in the context of differential equations is one where all terms are either zero or depend linearly on the variables, without constant additions. In this exercise, we aim to simplify our differential equations into a homogeneous form using a change of variables.

We introduce new variables \( x = w - R \) and \( y = p - R \), effectively centering our system around the room temperature \( R \). By substituting these into our original equations, we get:

• \( \frac{dx}{dt} = -k_m x \)
• \( \frac{dy}{dt} = -k_p (y - x) \)

These equations represent a homogeneous system because each equation contains only terms involving \( x \) or \( y \) without any standalone constant terms. This transformation significantly simplifies the process of finding solutions, as it removes the need to directly consider the external influence of room temperature on the system.

Initial-Value Problem

An initial-value problem involves solving a differential equation given specific starting values for the variables. For our cooling problem, initial values are specified as \( w(0) = w_0 \) and \( p(0) = p_0 \). These initial conditions are crucial because they define the starting point of our temperature changes and influence the specific solution we seek.

By incorporating these initial conditions, we can refine our homogeneous solutions and express them entirely in terms of \( w_0 \), \( p_0 \), and \( R \). The solutions become:

• \( w(t) = R + (w_0 - R)e^{-k_m t} \)
• \( p(t) = R + (p_0 - R - \frac{k_p}{k_p - k_m}(w_0 - R) + \frac{k_p}{k_p - k_m}(w_0 - R) e^{-k_m t}) e^{-k_p t} \)

Thus, the initial conditions allow us to map the specific cooling curve for both the glass of water and the coin from their initial state to their eventual convergence towards room temperature.

Cooling Rates

Cooling rates in this context refer to how quickly the temperatures of the glass of water and the coin approach equilibrium with their surroundings. Represented by the coefficients \( k_m \) and \( k_p \), these rates are pivotal in describing the dynamics of temperature changes.

The larger the value of \( k \), the faster the rate of cooling or warming.

• For the water, \( k_m \) determines how quickly it adjusts its temperature closer to the room temperature \( R \).
• For the coin, \( k_p \) measures how rapidly it aligns its temperature with that of the water.

These rates are essential for accurately predicting the behavior of the system over time, such as how long it will take for both the water and the coin to stabilize at room temperature. Understanding these rates provides critical insights into the rate of thermal energy exchange, which has broader applications in fields ranging from engineering to environmental science.