0005.qti

<item ident="C1-7062" title="C1 | Homogeneous first-order linear IVP | ver. 7062">
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  <presentation>
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        <div class="exercise-statement">
          <p>
            <strong>C1.</strong>
          </p>
          <p> Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. </p>
          <p style="text-align:center;">
            <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" alt="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" title="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" data-latex="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}"/>
          </p>
          <p>Then show how to verify that your particular solution is correct.</p>
        </div>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C1.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to find the general solution to the given ODE, and the particular solution to the given IVP. &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?-2%20%5C,%20%7By'%7D%20=%206%20%5C,%20%7By%7D%20,%5Chspace%7B1em%7D%20y%5Cbig(%20%5Clog%5Cleft(2%5Cright)%20%5Cbig)=%20%5Cfrac%7B3%7D%7B8%7D" alt="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" title="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}" data-latex="-2 \, {y'} = 6 \, {y} ,\hspace{1em} y\big( \log\left(2\right) \big)= \frac{3}{8}"&gt;
  &lt;/p&gt;
  &lt;p&gt;Then show how to verify that your particular solution is correct.&lt;/p&gt;
&lt;/div&gt;

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            <h4>Partial Answer:</h4>
            <p style="text-align:center;">
              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = k e^{\left(-3 \, t\right)}" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"/>
            </p>
            <p style="text-align:center;">
              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?{y} = 3 \, e^{\left(-3 \, t\right)}" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-3 \, t\right)}"/>
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  &lt;h4&gt;Partial Answer:&lt;/h4&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%20k%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = k e^{\left(-3 \, t\right)}" title="{y} = k e^{\left(-3 \, t\right)}" data-latex="{y} = k e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
  &lt;p style="text-align:center;"&gt;
    &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%7By%7D%20=%203%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-3 \, t\right)}" title="{y} = 3 \, e^{\left(-3 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-3 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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