0004.qti

<item ident="D4-5049" title="D4 | Using Laplace transforms to solve IVPs | ver. 5049">
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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?-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8" alt="-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8" title="-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8" data-latex="-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8"/>
          </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} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}}" alt="\frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}}" title="\frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}}" data-latex="\frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\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?-12%20%5C,%20%7By%7D%20+%2012%20%5C,%20%5Cdelta%5Cleft(t%20-%201%5Cright)%20=%20-3%20%5C,%20%7By''%7D%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%20-8" alt="-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8" title="-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8" data-latex="-12 \, {y} + 12 \, \delta\left(t - 1\right) = -3 \, {y''} \hspace{2em} y(0)= 0 , y'(0)= -8"&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-%204%7D%20=%20-%5Cfrac%7B1%7D%7B4%20%5C,%20%7B%5Cleft(s%20+%202%5Cright)%7D%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20%7B%5Cleft(s%20-%202%5Cright)%7D%7D" alt="\frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}}" title="\frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}}" data-latex="\frac{1}{s^{2} - 4} = -\frac{1}{4 \, {\left(s + 2\right)}} + \frac{1}{4 \, {\left(s - 2\right)}}"&gt;.&lt;/p&gt;
&lt;/div&gt;

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        <response_label ident="answer1" rshuffle="No"/>
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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{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}" alt="\mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}" title="\mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}" data-latex="\mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}"/>
            </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{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}" alt="\mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}" title="\mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}" data-latex="\mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}"/>
            </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} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" alt="{y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" title="{y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}"/>
            </p>
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        <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%7B4%20%5C,%20e%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20-%204%7D%20-%20%5Cfrac%7B8%7D%7Bs%5E%7B2%7D%20-%204%7D" alt="\mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}" title="\mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}" data-latex="\mathcal{L}\{y\}= -\frac{4 \, e^{\left(-s\right)}}{s^{2} - 4} - \frac{8}{s^{2} - 4}"&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%7Be%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%20+%202%7D%20-%20%5Cfrac%7Be%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%20-%202%7D%20+%20%5Cfrac%7B2%7D%7Bs%20+%202%7D%20-%20%5Cfrac%7B2%7D%7Bs%20-%202%7D" alt="\mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}" title="\mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}" data-latex="\mathcal{L}\{y\}= \frac{e^{\left(-s\right)}}{s + 2} - \frac{e^{\left(-s\right)}}{s - 2} + \frac{2}{s + 2} - \frac{2}{s - 2}"&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-e%5E%7B%5Cleft(2%20%5C,%20t%20-%202%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)%20+%20e%5E%7B%5Cleft(-2%20%5C,%20t%20+%202%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)%20-%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" title="{y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}" data-latex="{y} = -e^{\left(2 \, t - 2\right)} \mathrm{u}\left(t - 1\right) + e^{\left(-2 \, t + 2\right)} \mathrm{u}\left(t - 1\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{\left(-2 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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