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      <text>19.4.1.1 Modelling with first-order equations</text>
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      <text><![CDATA[<h3>Modelling with first-order equations<br></h3>
<p><br></p>
<h5>Applying Newton’s law of cooling<br></h5>

<p>In the <em>Modelling with Differential Equations</em> section we introduced Newton’s law of cooling. The model equation is</p>
<p>\( \qquad \dfrac{d\theta}{dt} = -k(\theta-\theta_\text{s}) \qquad\quad \theta=\theta_0\ \ \text{at}\ \ t=0. \qquad \qquad \qquad\) (5)</p>
<p>where \(\theta=\theta(t)\) is the temperature of the cooling object at time \(t, \theta_\text{s}\) the temperature of the environment (assumed constant) and \(k\) is a thermal constant related to the object, \(\theta_0\) is the initial temperature of
    the liquid.</p>
<p><br></p>
<hr>
<h4 class="HELM_example">Example</h4>
<p>Solve this initial value problem:</p>
<p>\( \qquad \dfrac{d\theta}{dt} = -k(\theta-\theta_s), \qquad \theta = \theta_0 \quad at\ \ \ t=0\)<br></p>

<h4 class="HELM_solution">Solution</h4>
<p>Separate the variables to obtain an equation connecting two integrals:</p>
<p>\(\qquad \int{\dfrac{d\theta}{\theta-\theta_\text{s}}} = -\int k \ dt\)</p>
<p>Now integrate both sides of this equation:</p>
<p>\( \qquad \ln (\theta-\theta_\text{s}) = -kt+C\) where \(C\) is constant.</p>
<p>Apply the initial condition and take exponentials to obtain a formula for \(\theta:\)</p>
<p>\( \qquad \ln (\theta_0-\theta_\text{s})=C.\)</p>
<p>Hence \(\quad \ln (\theta-\theta_\text{s}) = -kt+\ln (\theta_0-\theta_\text{s})\quad\) so that \(\qquad \ln (\theta-\theta_\text{s})-\ln (\theta_0-\theta_0) = -kt.\)</p>
<p>Thus, rearranging and inverting, we find:</p>
<p>\( \qquad \ln\left(\dfrac{\theta-\theta_\text{s}}{\theta_0-\theta_\text{s}} \right) = -kt\qquad \therefore \qquad \dfrac{\theta-\theta_\text{s}}{\theta_0-\theta_\text{s}} = \text{e}^{-kt} \quad\)</p>
<p>giving \(\quad \theta = \theta_\text{s} + (\theta_0 - \theta_\text{s}) \text{e}^{-kt}.\)
</p>
<p>The graph of \(\theta\) against \(t\) for \(\theta = \theta_s+(\theta_0-\theta_s) e^{-kt}\) is shown in the figure below.</p>

<figure class="HELM_figure">
  <img src="@@PLUGINFILE@@/fig1.svg" alt="Curve showing the drop of temperature over time. The initial temperature at t=0 is theta_zero and the drop is exponential." width="312" height="201" class="img-responsive atto_image_button_text-bottom">
</figure>

<p>We see that as time increases (\(t\to \infty\)), then the temperature of the object cools down to that of the environment, that is: \(\theta\to \theta_\text{s}.\)</p>
<p>We could have solved (5) by the integrating factor method, which you are now asked to do.</p>]]></text>
<file name="fig1.svg" path="/" encoding="base64">[encoded data, 18176 characters]</file>
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