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<h2 class="hd hd-2 unit-title">9.1. Recitation video: review using partial fractions</h2>
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<h3 class="hd hd-2">Partial fraction and inverse Laplace: problem setup</h3>
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<h3 class="hd hd-3 problem-header">Problem statement from recitation</h3><p>
Take a moment to work through the problems from the recitation video on your own. Once you are satisfied with your answer, watch the next video to see a worked solution. </p><ol class="enumerate"><li value="1"><p>
. Give the formula for [mathjaxinline]\mathcal{L}(f')[/mathjaxinline]. </p></li><li value="2"><p>
Find [mathjaxinline]\mathcal{L}^{-1}[/mathjaxinline] of each of the following:<br/>a) [mathjaxinline]\displaystyle \frac{1}{s^2-4}[/mathjaxinline] b) [mathjaxinline]\displaystyle \frac{s^2}{s^2+4}[/mathjaxinline] c) [mathjaxinline]\displaystyle \frac{e^{-5s}}{s^2-4}[/mathjaxinline] </p></li><li value="3"><p>
Write out the partial fraction decomposition for [mathjaxinline]\displaystyle {\frac{1}{s^2(s^2+4)(s+1)(s+3)}}[/mathjaxinline]. Do not solve for the coefficients, but give the Laplace inverse in terms of the letters you used for the coefficients. </p></li></ol><p>
If you need to, you should consult the Laplace table. </p><p><div class="hideshowbox"><h4 onclick="hideshow(this);" style="margin: 0px">Laplace table<span class="icon-caret-down toggleimage"/></h4><div class="hideshowcontent"><p><h3>Calculations</h3></p><table id="a0000001002" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001003"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle \displaystyle u(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}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.4)</td></tr><tr id="a0000001004"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>\mathrm{Re}\, r[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.5)</td></tr><tr id="a0000001005"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(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}{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.6)</td></tr><tr id="a0000001006"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(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{\omega }{s^2+\omega ^2}, \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.7)</td></tr><tr id="a0000001007"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s > 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.8)</td></tr><tr id="a0000001008"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s > 0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.9)</td></tr><tr id="a0000001009"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.10)</td></tr><tr id="a0000001010"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.11)</td></tr><tr id="a0000001011"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.12)</td></tr><tr id="a0000001012"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t)\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">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle \mathrm{Re}\, s>0[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.13)</td></tr></table><p><h3>Rules</h3></p><table id="a0000001013" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000001014"><td style="width:40%; border:none"> </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">
</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"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.14)</td></tr><tr id="a0000001015"><td style="width:40%; border:none"> </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">
</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"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.15)</td></tr><tr id="a0000001016"><td style="width:40%; border:none"> </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">
</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"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.16)</td></tr><tr id="a0000001017"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)f(t-a)[/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 e^{-as}F(s), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, first form}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.17)</td></tr><tr id="a0000001018"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">
[mathjaxinline]\displaystyle u(t-a)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 e^{-as}\mathcal{L}(f(t+a);s), \qquad \qquad[/mathjaxinline]
</td><td style="vertical-align:middle; text-align:center; border:none">
</td><td style="vertical-align:middle; text-align:left; border:none">
[mathjaxinline]\displaystyle t\text {-shift rule, second form}[/mathjaxinline]
</td><td style="width:40%; border:none"> </td><td style="width:20%; border:none;text-align:right" class="eqnnum">(6.18)</td></tr></table></div><p class="hideshowbottom" onclick="hideshow(this);" style="margin: 0px"><a href="javascript: {return false;}">Show</a></p></div></p><SCRIPT src="/assets/courseware/v1/631e447105fca1b243137b21b9ed6f90/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.js" type="text/javascript"/><LINK href="/assets/courseware/v1/daf81af0af57b85a105e0ed27b7873a0/asset-v1:OCW+18.031+2019_Spring+type@asset+block/latex2edx.css" rel="stylesheet" type="text/css"/>
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<h3 class="hd hd-2">Partial fraction and inverse Laplace: solution</h3>
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<h2 class="hd hd-2 unit-title">9.2. Solving ODEs</h2>
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<h3 class="hd hd-2">Recitation video: problem setup</h3>
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<h3 class="hd hd-3 problem-header">Problem statement from recitation</h3><p>
Take a moment to work through the problems from the recitation video on your own. Once you are satisfied with your answer, watch the next video to see a worked solution. </p><ol class="enumerate"><li value="1"><p>
a. Use the Laplace transform (and partial fractions) to solve the initial value problem (IVP) </p><table id="a0000001019" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]\dot x + 2x = 3\delta (t) + 5u(t), \quad x(0^-) = 0.[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table><p>
b. Give an IVP without any [mathjaxinline]\delta (t)[/mathjaxinline] in the input that has the same solution as in part (a). </p></li><li value="2"><p>
Solve [mathjaxinline]\ddot x + 9x = u(t), \quad x(0^-)=0, \dot x(0^-)=0.[/mathjaxinline] </p></li></ol><p><b class="bfseries"><span style="color:#FF7800">Note on video below:</span></b> In the video solution below to 1(b), David finds new pre-initial conditions for a homogeneous initial value problem that is consistent with the problems with delta function input. This should be a new <b class="bfseries"><span style="color:#0000FF">post-initial condition</span></b> rather than a pre-initial condition. </p>
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<h3 class="hd hd-2">Recitation video: solution</h3>
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<b class="bfseries"><span style="color:#FF7800">Note on video:</span></b> At the minute mark 9:30 in the video above, where it says that <blockquote class="quote"> "the function [mathjaxinline]x(t)[/mathjaxinline] starts growing continuously from 1 and then achieves an oscillation with period 3" </blockquote> should be <blockquote class="quote"> "the function x(t) starts growing continuously from 0 and then achieves an oscillation with period [mathjaxinline]2\pi /3[/mathjaxinline]." </blockquote>
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