<item ident="D4-7629" title="D4 | Using Laplace transforms to solve IVPs | ver. 7629">
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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?-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0" alt="-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0" title="-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0" data-latex="-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0"/>
</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} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}" alt="\frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}" title="\frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}" data-latex="\frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}"/>.</p>
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<mattext texttype="text/html"><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%20%5C,%20%7By%7D%20-%203%20%5C,%20%7By''%7D%20-%2012%20%5C,%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%202%5Cright)%20=%200%20%5Chspace%7B2em%7D%20y(0)=%205%20,%20y'(0)=%200" alt="-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0" title="-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0" data-latex="-12 \, {y} - 3 \, {y''} - 12 \, \mathrm{u}\left(t - 2\right) = 0 \hspace{2em} y(0)= 5 , y'(0)= 0">
</p>
<p>Hint: <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+%204%20%5C,%20s%7D%20=%20-%5Cfrac%7Bs%7D%7B4%20%5C,%20%7B%5Cleft(s%5E%7B2%7D%20+%204%5Cright)%7D%7D%20+%20%5Cfrac%7B1%7D%7B4%20%5C,%20s%7D" alt="\frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}" title="\frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}" data-latex="\frac{1}{s^{3} + 4 \, s} = -\frac{s}{4 \, {\left(s^{2} + 4\right)}} + \frac{1}{4 \, s}">.</p>
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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{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\right)} s}" alt="\mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\right)} s}" title="\mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\right)} s}" data-latex="\mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\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(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\right)}}{s}" alt="\mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\right)}}{s}" title="\mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\right)}}{s}" data-latex="\mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\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?{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)" alt="{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)" title="{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)" data-latex="{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)"/>
</p>
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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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20%5Cfrac%7B5%20%5C,%20s%7D%7Bs%5E%7B2%7D%20+%204%7D%20-%20%5Cfrac%7B4%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20s%5Cright)%7D%7D%7B%7B%5Cleft(s%5E%7B2%7D%20+%204%5Cright)%7D%20s%7D" alt="\mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\right)} s}" title="\mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\right)} s}" data-latex="\mathcal{L}\{y\}= \frac{5 \, s}{s^{2} + 4} - \frac{4 \, e^{\left(-2 \, s\right)}}{{\left(s^{2} + 4\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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20%5Cfrac%7Bs%20e%5E%7B%5Cleft(-2%20%5C,%20s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20+%204%7D%20+%20%5Cfrac%7B5%20%5C,%20s%7D%7Bs%5E%7B2%7D%20+%204%7D%20-%20%5Cfrac%7Be%5E%7B%5Cleft(-2%20%5C,%20s%5Cright)%7D%7D%7Bs%7D" alt="\mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\right)}}{s}" title="\mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\right)}}{s}" data-latex="\mathcal{L}\{y\}= \frac{s e^{\left(-2 \, s\right)}}{s^{2} + 4} + \frac{5 \, s}{s^{2} + 4} - \frac{e^{\left(-2 \, s\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?%7By%7D%20=%20%5Ccos%5Cleft(2%20%5C,%20t%20-%204%5Cright)%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%202%5Cright)%20+%205%20%5C,%20%5Ccos%5Cleft(2%20%5C,%20t%5Cright)%20-%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%202%5Cright)" alt="{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)" title="{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)" data-latex="{y} = \cos\left(2 \, t - 4\right) \mathrm{u}\left(t - 2\right) + 5 \, \cos\left(2 \, t\right) - \mathrm{u}\left(t - 2\right)">
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