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<item ident="D3-4847" title="D3 | Inverse Laplace transforms | ver. 4847">
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          <p>
            <strong>D3.</strong>
          </p>
          <p> Explain how to use convolution along with the formula <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\mathcal{L}\{e^{at})=\frac{1}{s-a}" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"/>to find the following inverse Laplace transform: </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}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}"/>
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      <mattext texttype="text/html">&lt;div class="exercise-statement"&gt;
  &lt;p&gt;
    &lt;strong&gt;D3.&lt;/strong&gt;
  &lt;/p&gt;
  &lt;p&gt; Explain how to use convolution along with the formula &lt;img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?%5Cmathcal%7BL%7D%5C%7Be%5E%7Bat%7D)=%5Cfrac%7B1%7D%7Bs-a%7D" alt="\mathcal{L}\{e^{at})=\frac{1}{s-a}" title="\mathcal{L}\{e^{at})=\frac{1}{s-a}" data-latex="\mathcal{L}\{e^{at})=\frac{1}{s-a}"&gt;to find the following inverse Laplace transform: &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%5E%7B-1%7D%5Cleft%5C%7B%20%5Cfrac%7B3%7D%7B2%20%5C,%20%7B%5Cleft(s%20+%208%5Cright)%7D%20%7B%5Cleft(s%20-%203%5Cright)%7D%7D%20%5Cright%5C%7D" alt="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" title="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}" data-latex="\mathcal{L}^{-1}\left\{ \frac{3}{2 \, {\left(s + 8\right)} {\left(s - 3\right)}} \right\}"&gt;
  &lt;/p&gt;
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            <h4>Partial Answer:</h4>
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              <img style="border:1px #ddd solid;padding:5px;border-radius:5px;" src="https://latex.codecogs.com/svg.latex?\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" alt="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" title="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}"/>
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  &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?%5Cfrac%7B3%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(3%20%5C,%20t%5Cright)%7D%20-%20%5Cfrac%7B3%7D%7B22%7D%20%5C,%20e%5E%7B%5Cleft(-8%20%5C,%20t%5Cright)%7D" alt="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" title="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}" data-latex="\frac{3}{22} \, e^{\left(3 \, t\right)} - \frac{3}{22} \, e^{\left(-8 \, t\right)}"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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