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      <text>19.2.9.4 First Order DE Ex3</text>
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      <text><![CDATA[<p class="HELM_exercise">Exercise</p>
<p>Find the general solution of the equation \(\dfrac{\mathrm{d}y}{\mathrm{d}t} + (\tan t)y = \cos t\).</p>
<p>\(y =\) [[input:ans1]] [[validation:ans1]] \(+ C\cos t\). [[feedback:prt1]] </p>]]></text>
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      <text><![CDATA[<p>We are given the equation \(\dfrac{\mathrm{d}y}{\mathrm{d}t} + (\tan t)y = \cos t\). This equation is already in standard form. The coefficient in front of \(y\) is \(\tan t\). So the integrating factor is \[e^{\int \tan t \ \mathrm{d}t} = e^{\int \frac{\sin t}{\cos t} \ \mathrm{d}t} = e^{-\log(\cos t)} = -\frac{1}{\cos t}.\]</p>

<p>Multiply the standard form of the equation by the integrating factor to get \[ \frac{1}{\cos t} \dfrac{\mathrm{d}y}{\mathrm{d}t} + \dfrac{\sin t}{\cos^2 t}y = 1.\] Recognise that this is now exact and construct an integral \[ \frac{\mathrm{d}}{\mathrm{d}t}\left(
    \frac{1}{\cos t}y \right) = 1 \iff \frac{1}{\cos t}y = \int 1 \ \mathrm{d}t.\] </p>

<p>Integrating gives \[ \frac{1}{\cos t}y = t + C\] which implies \[ y = t\cos t + C\cos t.\]</p>]]></text>
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      <text>ta1: t*cos(t);</text>
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