0000.qti

<item ident="D4-3628" title="D4 | Using Laplace transforms to solve IVPs | ver. 3628">
  <itemmetadata>
    <qtimetadata>
      <qtimetadatafield>
        <fieldlabel>question_type</fieldlabel>
        <fieldentry>essay_question</fieldentry>
      </qtimetadatafield>
    </qtimetadata>
  </itemmetadata>
  <presentation>
    <material>
      <mattextxml>
        <div class="exercise-statement">
          <p>
            <strong>D4.</strong>
          </p>
          <p> Explain how to solve the following 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} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2" alt="2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2" title="2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2" data-latex="2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2"/>
          </p>
          <p>Hint: <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}" alt="\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}" title="\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}" data-latex="\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}"/>.</p>
        </div>
      </mattextxml>
      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D4.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to solve the following 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=%20-2%20%5C,%20%7By''%7D%20-%202%20%5C,%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%20-2" alt="2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2" title="2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2" data-latex="2 \, {y} = -2 \, {y''} - 2 \, \mathrm{u}\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= -2"&gt;
  &lt;/p&gt;
  &lt;p&gt;Hint: &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cfrac%7B1%7D%7Bs%5E%7B3%7D%20+%20s%7D%20=%20-%5Cfrac%7Bs%7D%7Bs%5E%7B2%7D%20+%201%7D%20+%20%5Cfrac%7B1%7D%7Bs%7D" alt="\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}" title="\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}" data-latex="\frac{1}{s^{3} + s} = -\frac{s}{s^{2} + 1} + \frac{1}{s}"&gt;.&lt;/p&gt;
&lt;/div&gt;

</mattext>
    </material>
    <response_str ident="response1" rcardinality="Single">
      <render_fib>
        <response_label ident="answer1" rshuffle="No"/>
      </render_fib>
    </response_str>
  </presentation>
  <itemfeedback ident="general_fb">
    <flow_mat>
      <material>
        <mattextxml>
          <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?\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}" alt="\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}" title="\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}" data-latex="\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}"/>
            </p>
            <p style="text-align:center;">
              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}" alt="\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}" title="\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}" data-latex="\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}"/>
            </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} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right)" alt="{y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right)" title="{y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right)" data-latex="{y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20-%5Cfrac%7B2%7D%7Bs%5E%7B2%7D%20+%201%7D%20-%20%5Cfrac%7Be%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7B%7B%5Cleft(s%5E%7B2%7D%20+%201%5Cright)%7D%20s%7D" alt="\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}" title="\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}" data-latex="\mathcal{L}\{y\}= -\frac{2}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{{\left(s^{2} + 1\right)} s}"&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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20%5Cfrac%7Bs%20e%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20+%201%7D%20-%20%5Cfrac%7Be%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%7D%20-%20%5Cfrac%7B2%7D%7Bs%5E%7B2%7D%20+%201%7D" alt="\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}" title="\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}" data-latex="\mathcal{L}\{y\}= \frac{s e^{\left(-3 \, s\right)}}{s^{2} + 1} - \frac{e^{\left(-3 \, s\right)}}{s} - \frac{2}{s^{2} + 1}"&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=%20%5Ccos%5Cleft(t%20-%203%5Cright)%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(t%5Cright)%20-%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)" alt="{y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right)" title="{y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right)" data-latex="{y} = \cos\left(t - 3\right) \mathrm{u}\left(t - 3\right) - 2 \, \sin\left(t\right) - \mathrm{u}\left(t - 3\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

</mattext>
      </material>
    </flow_mat>
  </itemfeedback>
</item>