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<h2 class="hd hd-2 unit-title">PS2.1. Laplace transforms</h2>
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<h2 class="hd hd-2 unit-title">PS2.2. Solve the IVP</h2>
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Solve the IVP
</h3>
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<p>
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<h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4>
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<p>
<h3>Calculations</h3>
</p>
<table id="a0000000910" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000911">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle 1[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.115)</td>
</tr>
<tr id="a0000000912">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{rt}[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s-r}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;\mathrm{Re}\, r[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.116)</td>
</tr>
<tr id="a0000000913">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \cos \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.117)</td>
</tr>
<tr id="a0000000914">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \sin \omega t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.118)</td>
</tr>
<tr id="a0000000915">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac1{s^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.119)</td>
</tr>
<tr id="a0000000916">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t^ n[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{n!}{s^{n+1}}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s &gt; 0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.120)</td>
</tr>
<tr id="a0000000917">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.121)</td>
</tr>
<tr id="a0000000918">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle t\cos (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.122)</td>
</tr>
<tr id="a0000000919">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{2\omega }t\sin (\omega t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{s}{(s^2+\omega ^2)^2}, \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.123)</td>
</tr>
<tr id="a0000000920">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \frac{1}{2\omega ^2}\left(\frac{1}{\omega }\sin (\omega t)-t\cos (\omega t)\right)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \frac{1}{(s^2+\omega ^2)^2} , \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s&gt;0[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.124)</td>
</tr>
</table>
<p>
<h3>Rules</h3>
</p>
<table id="a0000000921" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto">
<tr id="a0000000922">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle f'(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle sF(s) - f(0), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.125)</td>
</tr>
<tr id="a0000000923">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle tf(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle -F'(s), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-derivative rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.126)</td>
</tr>
<tr id="a0000000924">
<td style="width:40%; border:none">&#160;</td>
<td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle e^{at}f(t)[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle \rightsquigarrow[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
[mathjaxinline]\displaystyle F(s-a), \qquad \qquad[/mathjaxinline]
</td>
<td style="vertical-align:middle; text-align:center; border:none">
&#160;
</td>
<td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle s\text {-shift rule}[/mathjaxinline]
</td>
<td style="width:40%; border:none">&#160;</td>
<td style="width:20%; border:none;text-align:right" class="eqnnum">(5.127)</td>
</tr>
</table>
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<p>
Solve [mathjaxinline]\ddot x+2\dot x+2x=\cos (2t)[/mathjaxinline] with initial condition [mathjaxinline]x(0)=1[/mathjaxinline], [mathjaxinline]\dot x(0)=1[/mathjaxinline], using Laplace transform. </p>
<p>
Find an expression for [mathjaxinline]X(s)[/mathjaxinline], the Laplace transform of [mathjaxinline]x(t)[/mathjaxinline]. <p style="display:inline">[mathjaxinline]X(s)=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps2-tab2-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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\(\)
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<p>
Find [mathjaxinline]x(t)[/mathjaxinline]. <p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_ps2-tab2-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p>
What initial conditions give a purely sinusoidal solution? </p>
<p>
<p style="display:inline">[mathjaxinline]x(0)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_ps2-tab2-problem1_4_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<p id="answer_ps2-tab2-problem1_4_1" class="answer"/>
<div id="input_ps2-tab2-problem1_4_1_preview" class="equation">
\(\)
<img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/>
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<div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/>
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<p>
<p style="display:inline">[mathjaxinline]\dot x(0)=[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_ps2-tab2-problem1_5_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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\(\)
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<h2 class="hd hd-2 unit-title">PS2.3. Another IVP problem</h2>
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Solve the IVP [mathjaxinline]\ddot{x} + 4x = t[/mathjaxinline] with initial condition [mathjaxinline]x(0) = 0[/mathjaxinline], [mathjaxinline]\dot{x}(0) = 2[/mathjaxinline], in two ways: (i) by combining homogeneous and particular solution (note: you can guess a particular solution of the form [mathjaxinline]f(t) = At + B[/mathjaxinline]); (ii) by using the Laplace transform and solving back. </p>
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Find an expression for [mathjaxinline]X(s)[/mathjaxinline], the Laplace transform of [mathjaxinline]x(t)[/mathjaxinline]. <p style="display:inline">[mathjaxinline]X(s)=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_ps2-tab3-problem1_2_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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Find [mathjaxinline]x(t)[/mathjaxinline]. <p style="display:inline">[mathjaxinline]x(t)=[/mathjaxinline]</p> <div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_ps2-tab3-problem1_3_1" class="inputtype formulaequationinput" style="display:inline-block;vertical-align:top">
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<h2 class="hd hd-2 unit-title">PS2.4. Inverse transform</h2>
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Inverse Laplace transform
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Use the [mathjaxinline]s[/mathjaxinline]-shift and [mathjaxinline]s[/mathjaxinline]-derivative rules to find: </p>
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<p style="display:inline">(a) [mathjaxinline]\mathcal{L}^{-1} \left( \frac{4}{(s-2)^2} \right)=[/mathjaxinline]</p>
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(b) [mathjaxinline]\mathcal{L}\left(t^ ne^{at} \right)[/mathjaxinline]. Check a special case in the answer box below. (Hint: type factorial(5) for 5!) </p>
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<p style="display:inline">[mathjaxinline]\mathcal{L}\left(t^{11}e^{at} \right)=[/mathjaxinline]</p>
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<p style="display:inline">(c) [mathjaxinline]\mathcal{L}\left( \frac{1}{2\omega ^2} \left(\sin (\omega t) - \omega t \cos (\omega t) \right)\right)=[/mathjaxinline]</p>
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<p style="display:inline">(d) [mathjaxinline]\mathcal{L}^{-1} \left(\frac{s^2}{\left(s^2+\omega ^2\right)^2} \right)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">PS2.5. Block diagram</h2>
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(This problem is not checked by the checker.) Sketch the block diagram for the system with the transfer function </p><table id="a0000000952" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]H(s) = \left( H_1(s)H_2(s)+ H_3(s) - H_4(s) \left( H_5(s)+H_6(s) \right) \right) H_7(s).[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table>
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<h2 class="hd hd-2 unit-title">PS2.6. Impedance</h2>
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Impedance
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The human circulatory system can be modeled as a power source with a resistor in parallel with a capacitor, where the resistor represents the capillary network and the capacitor represents large, compliant arteries. </p>
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(a) What is the impedance of this circuit? </p>
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(b) What is the transfer function and the gain if we consider the voltage across the power source to be in the input signal and the current through the circuit to be the system response? <i class="itshape">(This would correspond to the heart imposing a fixed pressure drop)</i>. </p>
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<p style="display:inline">Transfer function:</p>
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<p style="display:inline">Gain:</p>
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(c) What if the current through the circuit is the input signal and the voltage across the circuit is the system response? <i class="itshape">(This would correspond to the heart imposing a fixed flow)</i>. </p>
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<p style="display:inline">Transfer function:</p>
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<p style="display:inline">Gain:</p>
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(d) What is the sinusoidal response to the input signal [mathjaxinline]\cos (\omega t)[/mathjaxinline] for part (b)? </p>
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<p style="display:inline">System response:</p>
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