0003.qti

<item ident="D4-0666" title="D4 | Using Laplace transforms to solve IVPs | ver. 0666">
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          <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?0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6" alt="0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6" title="0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6" data-latex="0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6"/>
          </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} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}" alt="\frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}" title="\frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}" data-latex="\frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}"/>.</p>
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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?0%20=%203%20%5C,%20%7By''%7D%20+%2027%20%5C,%20%7By%7D%20+%2027%20%5C,%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%20-6" alt="0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6" title="0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6" data-latex="0 = 3 \, {y''} + 27 \, {y} + 27 \, \mathrm{u}\left(t - 1\right) \hspace{2em} y(0)= 0 , y'(0)= -6"&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+%209%20%5C,%20s%7D%20=%20-%5Cfrac%7Bs%7D%7B9%20%5C,%20%7B%5Cleft(s%5E%7B2%7D%20+%209%5Cright)%7D%7D%20+%20%5Cfrac%7B1%7D%7B9%20%5C,%20s%7D" alt="\frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}" title="\frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}" data-latex="\frac{1}{s^{3} + 9 \, s} = -\frac{s}{9 \, {\left(s^{2} + 9\right)}} + \frac{1}{9 \, s}"&gt;.&lt;/p&gt;
&lt;/div&gt;

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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{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\right)} s}" alt="\mathcal{L}\{y\}= -\frac{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\right)} s}" title="\mathcal{L}\{y\}= -\frac{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\right)} s}" data-latex="\mathcal{L}\{y\}= -\frac{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\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(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}" alt="\mathcal{L}\{y\}= \frac{s e^{\left(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}" title="\mathcal{L}\{y\}= \frac{s e^{\left(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}" data-latex="\mathcal{L}\{y\}= \frac{s e^{\left(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}"/>
            </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(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)" alt="{y} = \cos\left(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)" title="{y} = \cos\left(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)" data-latex="{y} = \cos\left(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)"/>
            </p>
          </div>
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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%7B6%7D%7Bs%5E%7B2%7D%20+%209%7D%20-%20%5Cfrac%7B9%20%5C,%20e%5E%7B%5Cleft(-s%5Cright)%7D%7D%7B%7B%5Cleft(s%5E%7B2%7D%20+%209%5Cright)%7D%20s%7D" alt="\mathcal{L}\{y\}= -\frac{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\right)} s}" title="\mathcal{L}\{y\}= -\frac{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\right)} s}" data-latex="\mathcal{L}\{y\}= -\frac{6}{s^{2} + 9} - \frac{9 \, e^{\left(-s\right)}}{{\left(s^{2} + 9\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(-s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20+%209%7D%20-%20%5Cfrac%7Be%5E%7B%5Cleft(-s%5Cright)%7D%7D%7Bs%7D%20-%20%5Cfrac%7B6%7D%7Bs%5E%7B2%7D%20+%209%7D" alt="\mathcal{L}\{y\}= \frac{s e^{\left(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}" title="\mathcal{L}\{y\}= \frac{s e^{\left(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}" data-latex="\mathcal{L}\{y\}= \frac{s e^{\left(-s\right)}}{s^{2} + 9} - \frac{e^{\left(-s\right)}}{s} - \frac{6}{s^{2} + 9}"&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(3%20%5C,%20t%20-%203%5Cright)%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)%20-%202%20%5C,%20%5Csin%5Cleft(3%20%5C,%20t%5Cright)%20-%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%201%5Cright)" alt="{y} = \cos\left(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)" title="{y} = \cos\left(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)" data-latex="{y} = \cos\left(3 \, t - 3\right) \mathrm{u}\left(t - 1\right) - 2 \, \sin\left(3 \, t\right) - \mathrm{u}\left(t - 1\right)"&gt;
  &lt;/p&gt;
&lt;/div&gt;

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