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Add modeling sections with real-world differential equation applications to all chapters #136

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167 changes: 167 additions & 0 deletions source/c0-whats-a-de/wad-model.ptx
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<!-- LICENSING
This file is part of the textbook "Exploring Differential Equations, An Interactive, Student-Centered Approach" by Geoffrey Cox.
License: Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)
You are free to:
• Share — copy and redistribute the material in any medium or format.
• Adapt — remix, transform, and build upon the material for any purpose, even commercially.
Under the following terms:
• Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made.
• ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license.
Full license text: https://creativecommons.org/licenses/by-sa/4.0/legalcode
Recommended attribution: "Exploring Differential Equations" by Geoffrey Cox, licensed CC BY-SA 4.0.
-->
<section label="wad-modeling">
<title>Modeling with Differential Equations</title>

<introduction>
<p>
Differential equations are more than just mathematical abstractions—they are the language we use to describe how things change in the real world. From population growth to cooling coffee, from falling objects to chemical reactions, differential equations provide a framework for understanding and predicting dynamic behavior.
</p>

<p>
In this section, we'll explore how to translate a real-world scenario into a differential equation. This process, called <term>mathematical modeling</term>, involves identifying variables, making assumptions, and using logic and physical principles to construct an equation that captures the essence of the problem.
</p>
</introduction>

<subsection>
<title>The Problem: Newton's Law of Cooling</title>

<p>
Imagine you've just poured yourself a hot cup of coffee at temperature <m>180^\circ F</m> and placed it on a table in a room where the temperature is <m>70^\circ F</m>. You know from experience that the coffee will cool down over time, but can we predict exactly how the temperature changes?
</p>

<p>
This scenario can be modeled using Newton's Law of Cooling, which states that the rate at which an object's temperature changes is proportional to the difference between its temperature and the ambient (room) temperature.
</p>
</subsection>

<subsection>
<title>Assumptions</title>

<p>
Before we can build our model, we need to make some simplifying assumptions:
</p>

<ul>
<li>The room temperature remains constant at <m>70^\circ F</m>.</li>
<li>The coffee is well-mixed, so its temperature is uniform throughout.</li>
<li>Heat loss occurs only through the exposed surface of the coffee.</li>
<li>The rate of cooling is proportional to the temperature difference between the coffee and the room.</li>
<li>External factors like wind or humidity are negligible.</li>
</ul>
</subsection>

<subsection>
<title>Building the Model</title>

<p>
Let's define our variables:
</p>

<ul>
<li><m>T(t)</m>: the temperature of the coffee at time <m>t</m> (in minutes)</li>
<li><m>T_{\text{room}}</m>: the constant room temperature (<m>70^\circ F</m>)</li>
<li><m>k</m>: a positive constant representing the cooling rate</li>
</ul>

<p>
The rate of temperature change is the derivative <m>\frac{dT}{dt}</m>. According to Newton's Law of Cooling, this rate is proportional to the temperature difference:
<me>
\frac{dT}{dt} = -k(T - T_{\text{room}})
</me>
</p>

<p>
The negative sign indicates that temperature <em>decreases</em> when <m>T > T_{\text{room}}</m>. For our specific problem:
<me>
\frac{dT}{dt} = -k(T - 70)
</me>
</p>

<p>
<term>Parameters:</term>
</p>
<ul>
<li><m>k > 0</m>: The cooling constant, which depends on properties like the surface area of the coffee, the material of the cup, and air circulation. Larger <m>k</m> means faster cooling.</li>
<li><m>T_{\text{room}} = 70</m>: The ambient temperature in degrees Fahrenheit.</li>
</ul>
</subsection>

<subsection>
<title>Data Collection Activity</title>

<p>
<term>Objective:</term> Estimate the cooling constant <m>k</m> by collecting real temperature data.
</p>

<p>
<term>Materials Needed:</term>
</p>
<ul>
<li>A cup of hot water (or coffee)</li>
<li>A thermometer (digital or analog)</li>
<li>A timer or stopwatch</li>
<li>A room-temperature environment</li>
</ul>

<p>
<term>Procedure:</term>
</p>
<ol>
<li>Heat water to approximately <m>180^\circ F</m> and pour it into a cup.</li>
<li>Measure and record the room temperature.</li>
<li>Record the initial temperature of the water at <m>t = 0</m>.</li>
<li>Measure and record the water temperature every 2 minutes for 20 minutes.</li>
<li>Create a table with columns: Time (min), Temperature (°F), Temperature Difference <m>(T - T_{\text{room}})</m>.</li>
</ol>

<p>
<term>Analysis Questions:</term>
</p>
<ol>
<li>Plot the temperature <m>T</m> versus time <m>t</m>. What pattern do you observe?</li>
<li>Plot the temperature difference <m>(T - T_{\text{room}})</m> versus time. Does this appear exponential?</li>
<li>Based on your data, estimate the cooling constant <m>k</m>. (Hint: Use the fact that <m>T(t) - T_{\text{room}} = (T_0 - T_{\text{room}})e^{-kt}</m>, which we'll learn to derive in later chapters.)</li>
<li>How would the cooling rate change if you used a metal cup instead of a ceramic one?</li>
<li>What happens to the temperature as <m>t \to \infty</m>? Does this match your intuition?</li>
</ol>
</subsection>

<subsection>
<title>Analytical Questions</title>

<p>
Work through these questions to deepen your understanding:
</p>

<ol>
<li>If the room temperature were <m>80^\circ F</m> instead of <m>70^\circ F</m>, how would this affect the differential equation?</li>
<li>What if the coffee were initially at <m>60^\circ F</m> (colder than the room)? Would it warm up? How would the differential equation change?</li>
<li>Explain why the parameter <m>k</m> must be positive. What would happen if <m>k</m> were negative?</li>
<li>Suppose you have two cups of coffee, one at <m>180^\circ F</m> and one at <m>160^\circ F</m>. Which one cools faster initially? Why?</li>
</ol>
</subsection>

<subsection>
<title>Final Report</title>

<p>
Prepare a brief report (1-2 pages) that includes:
</p>

<ol>
<li>A summary of the modeling process, including your assumptions and the derived differential equation.</li>
<li>Your experimental data presented in both table and graph form.</li>
<li>An estimate of the cooling constant <m>k</m> with an explanation of your estimation method.</li>
<li>A discussion of how well the model fits your data and any discrepancies you observed.</li>
<li>Suggestions for improving the model or the experiment.</li>
</ol>
</subsection>

<conclusion>
<p>
This modeling exercise demonstrates how differential equations arise naturally from real-world phenomena. By identifying variables, making reasonable assumptions, and applying physical principles, we translated a everyday experience into a mathematical model. In future chapters, we'll learn techniques to solve this and other differential equations analytically.
</p>
</conclusion>

</section>
227 changes: 227 additions & 0 deletions source/c1-classification/class-model.ptx
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<!-- LICENSING
This file is part of the textbook "Exploring Differential Equations, An Interactive, Student-Centered Approach" by Geoffrey Cox.
License: Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)
You are free to:
• Share — copy and redistribute the material in any medium or format.
• Adapt — remix, transform, and build upon the material for any purpose, even commercially.
Under the following terms:
• Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made.
• ShareAlike — If you remix, transform, or build upon the material, you must distribute your contributions under the same license.
Full license text: https://creativecommons.org/licenses/by-sa/4.0/legalcode
Recommended attribution: "Exploring Differential Equations" by Geoffrey Cox, licensed CC BY-SA 4.0.
-->
<section label="class-modeling">
<title>Modeling with Classification</title>

<introduction>
<p>
Understanding the order and linearity of a differential equation isn't just an academic exercise—it has practical implications for modeling real-world systems. The classification of a differential equation tells us about the complexity of the underlying phenomenon and guides us toward appropriate solution methods.
</p>

<p>
In this section, we'll explore how different types of differential equations arise from different physical contexts and learn to classify equations that model real systems.
</p>
</introduction>

<subsection>
<title>The Problem: Spring-Mass Systems</title>

<p>
Consider a mass attached to a spring, suspended vertically. When you pull the mass down and release it, it oscillates up and down. The question is: can we predict the position of the mass at any given time?
</p>

<p>
This seemingly simple system can be modeled with differential equations of different orders and linearity, depending on which physical effects we include.
</p>
</subsection>

<subsection>
<title>Assumptions and Model Variations</title>

<p>
<term>Basic Model (Linear, Second-Order):</term>
</p>

<p>
Let <m>y(t)</m> represent the displacement of the mass from its equilibrium position at time <m>t</m>. The simplest model assumes:
</p>
<ul>
<li>The spring force is proportional to displacement (Hooke's Law): <m>F_{\text{spring}} = -ky</m></li>
<li>There is no friction or air resistance</li>
<li>The mass <m>m</m> is constant</li>
</ul>

<p>
By Newton's second law (<m>F = ma</m>), where <m>a = \frac{d^2y}{dt^2}</m> is acceleration:
<me>
m\frac{d^2y}{dt^2} = -ky
</me>
or equivalently:
<me>
\frac{d^2y}{dt^2} + \frac{k}{m}y = 0
</me>
</p>

<p>
This is a <term>linear, homogeneous, second-order</term> differential equation.
</p>

<p>
<term>Enhanced Model (Linear, Second-Order with Damping):</term>
</p>

<p>
If we include air resistance or friction, which opposes motion proportionally to velocity:
</p>
<ul>
<li>Damping force: <m>F_{\text{damping}} = -c\frac{dy}{dt}</m></li>
</ul>

<p>
The equation becomes:
<me>
m\frac{d^2y}{dt^2} + c\frac{dy}{dt} + ky = 0
</me>
</p>

<p>
This is still <term>linear and second-order</term>, but now includes a first derivative term representing damping.
</p>

<p>
<term>Nonlinear Model (Second-Order):</term>
</p>

<p>
For large displacements, the spring may not obey Hooke's Law perfectly. A more realistic spring force might be:
<me>
F_{\text{spring}} = -ky - \alpha y^3
</me>
</p>

<p>
This leads to a <term>nonlinear</term> differential equation:
<me>
m\frac{d^2y}{dt^2} + c\frac{dy}{dt} + ky + \alpha y^3 = 0
</me>
</p>

<p>
<term>Parameters:</term>
</p>
<ul>
<li><m>m</m>: mass (kg)</li>
<li><m>k</m>: spring constant (N/m), representing spring stiffness</li>
<li><m>c</m>: damping coefficient (N·s/m)</li>
<li><m>\alpha</m>: nonlinear spring coefficient</li>
</ul>
</subsection>

<subsection>
<title>Classification Activity</title>

<p>
<term>Objective:</term> Practice classifying differential equations from various physical contexts.
</p>

<p>
<term>Part 1: Classify These Equations</term>
</p>

<p>
For each differential equation below, determine:
</p>
<ol>
<li>The order</li>
<li>Whether it is linear or nonlinear</li>
<li>If linear, whether it is homogeneous or nonhomogeneous</li>
</ol>

<p>
<term>Equations to Classify:</term>
</p>
<ol>
<li>Population growth: <m>\frac{dP}{dt} = rP</m></li>
<li>Radioactive decay: <m>\frac{dN}{dt} = -\lambda N</m></li>
<li>Logistic growth: <m>\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)</m></li>
<li>Forced harmonic oscillator: <m>m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F_0\cos(\omega t)</m></li>
<li>Pendulum (small angles): <m>\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0</m></li>
<li>Pendulum (large angles): <m>\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0</m></li>
<li>RC circuit: <m>RC\frac{dV}{dt} + V = V_{\text{source}}(t)</m></li>
</ol>

<p>
<term>Part 2: Modeling Exercise</term>
</p>

<p>
Working in groups, construct a simple spring-mass system:
</p>

<p>
<term>Materials:</term>
</p>
<ul>
<li>A spring</li>
<li>Various masses (washers, weights, etc.)</li>
<li>A ruler or measuring tape</li>
<li>A timer</li>
</ul>

<p>
<term>Procedure:</term>
</p>
<ol>
<li>Hang the spring vertically and measure its natural (unstretched) length.</li>
<li>Attach a known mass and measure the new equilibrium length. The difference gives the stretch <m>\Delta y</m>.</li>
<li>Use <m>mg = k\Delta y</m> to estimate the spring constant <m>k</m>.</li>
<li>Pull the mass down slightly and release it. Time 10 complete oscillations and calculate the period <m>T</m>.</li>
<li>Repeat with different masses.</li>
</ol>

<p>
<term>Data Analysis:</term>
</p>
<ol>
<li>Create a table with mass, stretch, and period for each trial.</li>
<li>Calculate the spring constant <m>k</m> from each mass-stretch measurement.</li>
<li>Plot the period <m>T</m> versus mass <m>m</m>. What relationship do you observe?</li>
<li>The theoretical period for a spring-mass system is <m>T = 2\pi\sqrt{\frac{m}{k}}</m>. Does your data support this?</li>
</ol>
</subsection>

<subsection>
<title>Analytical Questions</title>

<ol>
<li>Why is the basic spring-mass equation second-order? What physical quantity is represented by the second derivative?</li>
<li>How does adding damping change the classification? Does it change the order or linearity?</li>
<li>What makes the large-angle pendulum equation nonlinear? How might this affect its behavior compared to the small-angle case?</li>
<li>If you wanted to model a spring-mass system on the surface of the Moon (where gravity is weaker), which parameters would change?</li>
<li>Can you think of other physical systems that might be modeled by second-order differential equations?</li>
</ol>
</subsection>

<subsection>
<title>Final Report</title>

<p>
Write a comprehensive report (2-3 pages) that includes:
</p>

<ol>
<li>A comparison of the three spring-mass models (basic, damped, nonlinear), discussing how assumptions affect the classification.</li>
<li>Your experimental data, including calculations of the spring constant and analysis of the period-mass relationship.</li>
<li>Answers to the classification questions in Part 1 of the activity, with detailed reasoning.</li>
<li>A discussion of when a linear model might be sufficient versus when a nonlinear model is necessary.</li>
<li>Real-world examples of systems modeled by equations of different orders and linearity.</li>
</ol>
</subsection>

<conclusion>
<p>
Classification is the first step in understanding a differential equation. By recognizing whether an equation is first-order or second-order, linear or nonlinear, we can anticipate its behavior and select appropriate solution techniques. The spring-mass system demonstrates how the same physical scenario can yield different equations depending on our modeling assumptions—and how those choices affect the mathematical classification.
</p>
</conclusion>

</section>
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