<item ident="D4-7074" title="D4 | Using Laplace transforms to solve IVPs | ver. 7074">
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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?-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2" alt="-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2" title="-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2" data-latex="-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2"/>
</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 - 1} + \frac{1}{s - 2}" alt="\frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2}" title="\frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2}" data-latex="\frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2}"/>.</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?-3%20%5C,%20%7By''%7D%20=%206%20%5C,%20%7By%7D%20-%209%20%5C,%20%7By'%7D%20-%203%20%5C,%20%5Cdelta%5Cleft(t%20-%202%5Cright)%20%5Chspace%7B2em%7D%20y(0)=%200%20,%20y'(0)=%20-2" alt="-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2" title="-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2" data-latex="-3 \, {y''} = 6 \, {y} - 9 \, {y'} - 3 \, \delta\left(t - 2\right) \hspace{2em} y(0)= 0 , y'(0)= -2">
</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-%201%7D%20+%20%5Cfrac%7B1%7D%7Bs%20-%202%7D" alt="\frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2}" title="\frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2}" data-latex="\frac{1}{s^{2} - 3 \, s + 2} = -\frac{1}{s - 1} + \frac{1}{s - 2}">.</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{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{s^{2} - 3 \, s + 2}" alt="\mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{s^{2} - 3 \, s + 2}" title="\mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{s^{2} - 3 \, s + 2}" data-latex="\mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{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{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \frac{2}{s - 2}" alt="\mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \frac{2}{s - 2}" title="\mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \frac{2}{s - 2}" data-latex="\mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \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 - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" alt="{y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" title="{y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" data-latex="{y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}"/>
</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%7Be%5E%7B%5Cleft(-2%20%5C,%20s%5Cright)%7D%7D%7Bs%5E%7B2%7D%20-%203%20%5C,%20s%20+%202%7D%20-%20%5Cfrac%7B2%7D%7Bs%5E%7B2%7D%20-%203%20%5C,%20s%20+%202%7D" alt="\mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{s^{2} - 3 \, s + 2}" title="\mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{s^{2} - 3 \, s + 2}" data-latex="\mathcal{L}\{y\}= \frac{e^{\left(-2 \, s\right)}}{s^{2} - 3 \, s + 2} - \frac{2}{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%7Be%5E%7B%5Cleft(-2%20%5C,%20s%5Cright)%7D%7D%7Bs%20-%201%7D%20+%20%5Cfrac%7Be%5E%7B%5Cleft(-2%20%5C,%20s%5Cright)%7D%7D%7Bs%20-%202%7D%20+%20%5Cfrac%7B2%7D%7Bs%20-%201%7D%20-%20%5Cfrac%7B2%7D%7Bs%20-%202%7D" alt="\mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \frac{2}{s - 2}" title="\mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \frac{2}{s - 2}" data-latex="\mathcal{L}\{y\}= -\frac{e^{\left(-2 \, s\right)}}{s - 1} + \frac{e^{\left(-2 \, s\right)}}{s - 2} + \frac{2}{s - 1} - \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?%7By%7D%20=%20e%5E%7B%5Cleft(2%20%5C,%20t%20-%204%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%202%5Cright)%20-%20e%5E%7B%5Cleft(t%20-%202%5Cright)%7D%20%5Cmathrm%7Bu%7D%5Cleft(t%20-%202%5Cright)%20-%202%20%5C,%20e%5E%7B%5Cleft(2%20%5C,%20t%5Cright)%7D%20+%202%20%5C,%20e%5E%7Bt%7D" alt="{y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" title="{y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}" data-latex="{y} = e^{\left(2 \, t - 4\right)} \mathrm{u}\left(t - 2\right) - e^{\left(t - 2\right)} \mathrm{u}\left(t - 2\right) - 2 \, e^{\left(2 \, t\right)} + 2 \, e^{t}">
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