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      <text>19.4.1.3 Integrating factor</text>
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      <text><![CDATA[<p>Multiplying (6) by this factor we find that</p>
<p>\(\qquad \text{e}^{kt} \dfrac{d\theta}{dt} + k\text{e}^{kt} \theta = k \theta_\text{s}\text{e}^{kt} \,\) or, rearranging, \( \quad \dfrac{d}{dt} (\text{e}^{kt}\theta) = k\theta_\text{s}\text{e}^{kt}.\)
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<p>Now integrate this equation and apply the initial condition:</p>
<p>Integration produces: \( \quad \text{e}^{kt}\theta = \theta_\text{s} \text{e}^{kt}+C\), where \(C\) is an arbitrary constant.</p>
<p>Then, applying the initial condition: when \(t=0, \, \theta_0 = \theta_\text{s} +C\) so that \(C=\theta_0-\theta_\text{s}\) gives the same result as before:</p>
<p>\(\qquad \theta = \theta_\text{s} + (\theta_0-\theta_\text{s}) \text{e}^{-kt}.\)</p>]]></text>
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