0005.qti

<item ident="D4-2484" title="D4 | Using Laplace transforms to solve IVPs | ver. 2484">
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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?-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4" alt="-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4" title="-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4" data-latex="-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4"/>
          </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} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}" alt="\frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}" title="\frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}" data-latex="\frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}"/>.</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?-3%20%5C,%20%7By''%7D%20=%2018%20%5C,%20%7By%7D%20-%2015%20%5C,%20%7By'%7D%20-%206%20%5C,%20%5Cdelta%5Cleft(t%20-%203%5Cright)%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%204" alt="-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4" title="-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4" data-latex="-3 \, {y''} = 18 \, {y} - 15 \, {y'} - 6 \, \delta\left(t - 3\right) \hspace{2em} y(0)= 0 , y'(0)= 4"&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-%205%20%5C,%20s%20+%206%7D%20=%20-%5Cfrac%7B1%7D%7Bs%20-%202%7D%20+%20%5Cfrac%7B1%7D%7Bs%20-%203%7D" alt="\frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}" title="\frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}" data-latex="\frac{1}{s^{2} - 5 \, s + 6} = -\frac{1}{s - 2} + \frac{1}{s - 3}"&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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            <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 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}" alt="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}" title="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}" data-latex="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}"/>
            </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(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \frac{4}{s - 3}" alt="\mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \frac{4}{s - 3}" title="\mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \frac{4}{s - 3}" data-latex="\mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \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 - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" alt="{y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, 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%7B2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20-%205%20%5C,%20s%20+%206%7D%20+%20%5Cfrac%7B4%7D%7Bs%5E%7B2%7D%20-%205%20%5C,%20s%20+%206%7D" alt="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}" title="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}" data-latex="\mathcal{L}\{y\}= \frac{2 \, e^{\left(-3 \, s\right)}}{s^{2} - 5 \, s + 6} + \frac{4}{s^{2} - 5 \, s + 6}"&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(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%20-%202%7D%20+%20%5Cfrac%7B2%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%20-%203%7D%20-%20%5Cfrac%7B4%7D%7Bs%20-%202%7D%20+%20%5Cfrac%7B4%7D%7Bs%20-%203%7D" alt="\mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \frac{4}{s - 3}" title="\mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \frac{4}{s - 3}" data-latex="\mathcal{L}\{y\}= -\frac{2 \, e^{\left(-3 \, s\right)}}{s - 2} + \frac{2 \, e^{\left(-3 \, s\right)}}{s - 3} - \frac{4}{s - 2} + \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=%202%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%20-%209%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)%20-%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%20-%206%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)%20+%204%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%204%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D" alt="{y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" title="{y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)}" data-latex="{y} = 2 \, e^{\left(3 \, t - 9\right)} \mathrm{u}\left(t - 3\right) - 2 \, e^{\left(2 \, t - 6\right)} \mathrm{u}\left(t - 3\right) + 4 \, e^{\left(3 \, t\right)} - 4 \, e^{\left(2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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