0010.qti

<item ident="D4-8829" title="D4 | Using Laplace transforms to solve IVPs | ver. 8829">
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  <presentation>
    <material>
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        <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?24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16" alt="24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16" title="24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16" data-latex="24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16"/>
          </p>
          <p>Hint: <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}" alt="\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}" title="\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}" data-latex="\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}"/>.</p>
        </div>
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      <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?24%20%5C,%20%5Cdelta%5Cleft(t%20-%201%5Cright)%20=%20-3%20%5C,%20%7By''%7D%20+%209%20%5C,%20%7By%7D%20+%206%20%5C,%20%7By'%7D%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%20-16" alt="24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16" title="24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16" data-latex="24 \, \delta\left(t - 1\right) = -3 \, {y''} + 9 \, {y} + 6 \, {y'} \hspace{2em} y(0)= 0 , y'(0)= -16"&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%7B2%7D%20-%202%20%5C,%20s%20-%203%7D%20=%20-%5Cfrac%7B1%7D%7B4%20%5C,%20%7B%5Cleft(s%20+%201%5Cright)%7D%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20%7B%5Cleft(s%20-%203%5Cright)%7D%7D" alt="\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}" title="\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}" data-latex="\frac{1}{s^{2} - 2 \, s - 3} = -\frac{1}{4 \, {\left(s + 1\right)}} + \frac{1}{4 \, {\left(s - 3\right)}}"&gt;.&lt;/p&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?\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}" alt="\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}" title="\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}" data-latex="\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}"/>
            </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{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}" alt="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}" title="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}" data-latex="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}"/>
            </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} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" alt="{y} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{y} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, 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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20-%5Cfrac%7B8%20%5C,%20e%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20-%202%20%5C,%20s%20-%203%7D%20-%20%5Cfrac%7B16%7D%7Bs%5E%7B2%7D%20-%202%20%5C,%20s%20-%203%7D" alt="\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}" title="\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}" data-latex="\mathcal{L}\{y\}= -\frac{8 \, e^{\left(-s\right)}}{s^{2} - 2 \, s - 3} - \frac{16}{s^{2} - 2 \, s - 3}"&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%7B2%20%5C,%20e%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%20+%201%7D%20-%20%5Cfrac%7B2%20%5C,%20e%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%20-%203%7D%20+%20%5Cfrac%7B4%7D%7Bs%20+%201%7D%20-%20%5Cfrac%7B4%7D%7Bs%20-%203%7D" alt="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}" title="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}" data-latex="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-s\right)}}{s + 1} - \frac{2 \, e^{\left(-s\right)}}{s - 3} + \frac{4}{s + 1} - \frac{4}{s - 3}"&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=%20-2%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%20-%203%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)%20+%202%20%5C,%20e%5E%7B%5Cleft(-t%20+%201%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)%20-%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20+%204%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D" alt="{y} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" title="{y} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}" data-latex="{y} = -2 \, e^{\left(3 \, t - 3\right)} \mathrm{u}\left(t - 1\right) + 2 \, e^{\left(-t + 1\right)} \mathrm{u}\left(t - 1\right) - 4 \, e^{\left(3 \, t\right)} + 4 \, e^{\left(-t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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