<item ident="D4-1525" title="D4 | Using Laplace transforms to solve IVPs | ver. 1525">
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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?4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3" alt="4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3" title="4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3" data-latex="4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3"/>
</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} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}" alt="\frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}" title="\frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}" data-latex="\frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}"/>.</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?4%20%5C,%20%7By%7D%20+%202%20%5C,%20%7By''%7D%20+%206%20%5C,%20%7By'%7D%20-%206%20%5C,%20%5Cdelta%5Cleft(t%20-%203%5Cright)%20=%200%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%203" alt="4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3" title="4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3" data-latex="4 \, {y} + 2 \, {y''} + 6 \, {y'} - 6 \, \delta\left(t - 3\right) = 0 \hspace{2em} y(0)= 0 , y'(0)= 3">
</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%7B2%7D%20+%203%20%5C,%20s%20+%202%7D%20=%20-%5Cfrac%7B1%7D%7Bs%20+%202%7D%20+%20%5Cfrac%7B1%7D%7Bs%20+%201%7D" alt="\frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}" title="\frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}" data-latex="\frac{1}{s^{2} + 3 \, s + 2} = -\frac{1}{s + 2} + \frac{1}{s + 1}">.</p>
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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{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, s + 2}" alt="\mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, s + 2}" title="\mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, s + 2}" data-latex="\mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, 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?\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}" alt="\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}" title="\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}" data-latex="\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}"/>
</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} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}" alt="{y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}"/>
</p>
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<mattext texttype="text/html"><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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20%5Cfrac%7B3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20+%203%20%5C,%20s%20+%202%7D%20+%20%5Cfrac%7B3%7D%7Bs%5E%7B2%7D%20+%203%20%5C,%20s%20+%202%7D" alt="\mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, s + 2}" title="\mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, s + 2}" data-latex="\mathcal{L}\{y\}= \frac{3 \, e^{\left(-3 \, s\right)}}{s^{2} + 3 \, s + 2} + \frac{3}{s^{2} + 3 \, 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?%5Cmathcal%7BL%7D%5C%7By%5C%7D=%20-%5Cfrac%7B3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%20+%202%7D%20+%20%5Cfrac%7B3%20%5C,%20e%5E%7B%5Cleft(-3%20%5C,%20s%5Cright)%7D%7D%7Bs%20+%201%7D%20-%20%5Cfrac%7B3%7D%7Bs%20+%202%7D%20+%20%5Cfrac%7B3%7D%7Bs%20+%201%7D" alt="\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}" title="\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}" data-latex="\mathcal{L}\{y\}= -\frac{3 \, e^{\left(-3 \, s\right)}}{s + 2} + \frac{3 \, e^{\left(-3 \, s\right)}}{s + 1} - \frac{3}{s + 2} + \frac{3}{s + 1}">
</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=%203%20%5C,%20e%5E%7B%5Cleft(-t%20+%203%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%20+%206%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%203%5Cright)%20+%203%20%5C,%20e%5E%7B%5Cleft(-t%5Cright)%7D%20-%203%20%5C,%20e%5E%7B%5Cleft(-2%20%5C,%20t%5Cright)%7D" alt="{y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}" title="{y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}" data-latex="{y} = 3 \, e^{\left(-t + 3\right)} \mathrm{u}\left(t - 3\right) - 3 \, e^{\left(-2 \, t + 6\right)} \mathrm{u}\left(t - 3\right) + 3 \, e^{\left(-t\right)} - 3 \, e^{\left(-2 \, t\right)}">
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