0005.qti

<item ident="C3-3912" title="C3 | Homogeneous second-order linear ODE | ver. 3912">
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  <presentation>
    <material>
      <mattextxml>
        <div class="exercise-statement">
          <p>
            <strong>C3.</strong>
          </p>
          <p>Explain how to find the general solution to each given ODE using exponential functions.</p>
          <p>For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.</p>
          <ol type="a">
            <li>
              <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''} = -4 \, {y'} - 2 \, {y}" alt="2 \, {y''} = -4 \, {y'} - 2 \, {y}" title="2 \, {y''} = -4 \, {y'} - 2 \, {y}" data-latex="2 \, {y''} = -4 \, {y'} - 2 \, {y}"/>
              </p>
            </li>
            <li>
              <p style="text-align:center;">
                <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?24 \, {x} = -12 \, {x'} - 3 \, {x''}" alt="24 \, {x} = -12 \, {x'} - 3 \, {x''}" title="24 \, {x} = -12 \, {x'} - 3 \, {x''}" data-latex="24 \, {x} = -12 \, {x'} - 3 \, {x''}"/>
              </p>
            </li>
          </ol>
        </div>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;C3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt;Explain how to find the general solution to each given ODE using exponential functions.&lt;/p&gt;
  &lt;p&gt;For each exponential solution using complex numbers, also provide an alternate general solution using only real numbers.&lt;/p&gt;
  &lt;ol type="a"&gt;
    &lt;li&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=%20-4%20%5C,%20%7By'%7D%20-%202%20%5C,%20%7By%7D" alt="2 \, {y''} = -4 \, {y'} - 2 \, {y}" title="2 \, {y''} = -4 \, {y'} - 2 \, {y}" data-latex="2 \, {y''} = -4 \, {y'} - 2 \, {y}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
    &lt;li&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?24%20%5C,%20%7Bx%7D%20=%20-12%20%5C,%20%7Bx'%7D%20-%203%20%5C,%20%7Bx''%7D" alt="24 \, {x} = -12 \, {x'} - 3 \, {x''}" title="24 \, {x} = -12 \, {x'} - 3 \, {x''}" data-latex="24 \, {x} = -12 \, {x'} - 3 \, {x''}"&gt;
      &lt;/p&gt;
    &lt;/li&gt;
  &lt;/ol&gt;
&lt;/div&gt;

</mattext>
    </material>
    <response_str ident="response1" rcardinality="Single">
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        <response_label ident="answer1" rshuffle="No"/>
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  <itemfeedback ident="general_fb">
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          <div class="exercise-answer">
            <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?{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, 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?{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, 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} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"/>
            </p>
          </div>
        </mattextxml>
        <mattext texttype="text/html">&lt;div class="exercise-answer"&gt;
  &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?%7Bx%7D%20=%20c_%7B1%7D%20e%5E%7B%5Cleft(%5Cleft(2%20i%20-%202%5Cright)%20%5C,%20t%5Cright)%7D%20+%20c_%7B2%7D%20e%5E%7B%5Cleft(-%5Cleft(2%20i%20+%202%5Cright)%20%5C,%20t%5Cright)%7D" alt="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" title="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, t\right)}" data-latex="{x} = c_{1} e^{\left(\left(2 i - 2\right) \, t\right)} + c_{2} e^{\left(-\left(2 i + 2\right) \, 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?%7Bx%7D%20=%20%7B%5Cleft(d_%7B1%7D%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20+%20d_%7B2%7D%20%5Csin%5Cleft(2%20%5C,%20t%5Cright)%5Cright)%7D%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" title="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, t\right)}" data-latex="{x} = {\left(d_{1} \cos\left(2 \, t\right) + d_{2} \sin\left(2 \, t\right)\right)} e^{\left(-2 \, 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=%20k_%7B1%7D%20t%20e%5E%7B%5Cleft(-t%5Cright)%7D%20+%20k_%7B2%7D%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" title="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}" data-latex="{y} = k_{1} t e^{\left(-t\right)} + k_{2} e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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