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<h2 class="hd hd-2 unit-title">Pset 3 checker</h2>
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<p><a href="/assets/courseware/v1/fbbd3e607fb1075985da373ea141fb0f/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/mites_1803_s24_ps3.pdf" target="[object Object]">Problem set 3 (PDF)</a></p>
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<h3 class="hd hd-3 problem-header" id="ps3-1-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-1-check-problem-progress" tabindex="-1">
Problem 1
</h3>
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<p>
To check your answers put them in the appropriate box and click the &#8216;Check' button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. </p>
<p>
Unless otherwise stated the checker expects numerical answers to 3 significant digits. </p>
<p>
For formulas you should use Matlab notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] <b class="bf">NOT </b> [mathjaxinline]3x[/mathjaxinline]. </p>
<p>
The checker knows many of the standard functions. Use exp(x) and ln(x) for the exponential and natural log functions. </p>
<p>
The formula checker expects a specific letter for the variables, so be sure to use the one specified. </p>
<p><b class="bf">1(a)</b> Give the particular solution found using the ERF as a function of [mathjaxinline]t[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]x_ p(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps3-1-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps3-1-check_2_1" id="input_ps3-1-check_2_1" aria-describedby="status_ps3-1-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps3-1-check_2_1"/>
<span class="status unanswered" id="status_ps3-1-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps3-1-check_2_1" class="answer"/>
<div id="display_ps3-1-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-1-check_2_1_dynamath" name="input_ps3-1-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">1(b)</b> Give the particular solution found using the ERF as a function of [mathjaxinline]t[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]x_ p(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps3-1-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps3-1-check_3_1" id="input_ps3-1-check_3_1" aria-describedby="status_ps3-1-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps3-1-check_3_1"/>
<span class="status unanswered" id="status_ps3-1-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps3-1-check_3_1" class="answer"/>
<div id="display_ps3-1-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-1-check_3_1_dynamath" name="input_ps3-1-check_3_1_dynamath"/>
</div>
</div></div>
</p>
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data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-2a-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-2a-check/handler/xmodule_handler"
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<h3 class="hd hd-3 problem-header" id="ps3-2a-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-2a-check-problem-progress" tabindex="-1">
Problem 2a
</h3>
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<div class="problem">
<div>
<p><b class="bf">2(a)</b> Give the periodic solution. Use letters m, b, k, B, w. For [mathjaxinline]x_ p[/mathjaxinline], you should spell out &#8216;phi' for the phase angle. </p>
<p>
<p style="display:inline">[mathjaxinline]x_ p(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps3-2a-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps3-2a-check_2_1" id="input_ps3-2a-check_2_1" aria-describedby="status_ps3-2a-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps3-2a-check_2_1"/>
<span class="status unanswered" id="status_ps3-2a-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps3-2a-check_2_1" class="answer"/>
<div id="display_ps3-2a-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-2a-check_2_1_dynamath" name="input_ps3-2a-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
Use the function arctan() to give a formula for [mathjaxinline]\phi[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\phi (\omega ) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps3-2a-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps3-2a-check_3_1" id="input_ps3-2a-check_3_1" aria-describedby="status_ps3-2a-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps3-2a-check_3_1"/>
<span class="status unanswered" id="status_ps3-2a-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps3-2a-check_3_1" class="answer"/>
<div id="display_ps3-2a-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-2a-check_3_1_dynamath" name="input_ps3-2a-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">Note:</b> The answer checker doesn't ask for which quadrants [mathjaxinline]\phi[/mathjaxinline] might be in, but you need to specify this on your paper. </p>
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<h3 class="hd hd-3 problem-header" id="ps3-2b-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-2b-check-problem-progress" tabindex="-1">
Problem 2b
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-2b-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">2(b)</b> Give the periodic solution. Use letters m, b, k, B, w. For [mathjaxinline]x_ p[/mathjaxinline], you should spell out &#8216;phi' for the phase angle. </p>
<p>
<p style="display:inline">[mathjaxinline]x_ p(t) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps3-2b-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps3-2b-check_2_1" id="input_ps3-2b-check_2_1" aria-describedby="status_ps3-2b-check_2_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps3-2b-check_2_1"/>
<span class="status unanswered" id="status_ps3-2b-check_2_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps3-2b-check_2_1" class="answer"/>
<div id="display_ps3-2b-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-2b-check_2_1_dynamath" name="input_ps3-2b-check_2_1_dynamath"/>
</div>
</div></div>
</p>
<p>
Use the function arctan() to give a formula for [mathjaxinline]\phi[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\phi (\omega ) =[/mathjaxinline]</p>
<div class="inline" tabindex="-1" aria-label="Question 2" role="group"><div id="inputtype_ps3-2b-check_3_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
<div class="script_placeholder" data-src="/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js"/>
<div class="unanswered inline">
<input type="text" name="input_ps3-2b-check_3_1" id="input_ps3-2b-check_3_1" aria-describedby="status_ps3-2b-check_3_1" value="" class="math" size="50"/>
<span class="trailing_text" id="trailing_text_ps3-2b-check_3_1"/>
<span class="status unanswered" id="status_ps3-2b-check_3_1" data-tooltip="Not yet answered.">
<span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/>
</span>
<p id="answer_ps3-2b-check_3_1" class="answer"/>
<div id="display_ps3-2b-check_3_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-2b-check_3_1_dynamath" name="input_ps3-2b-check_3_1_dynamath"/>
</div>
</div></div>
</p>
<p><b class="bf">Note:</b> The answer checker doesn't ask for which quadrants [mathjaxinline]\phi[/mathjaxinline] might be in, but you need to specify this on your paper. </p>
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Problem 2c
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<p><b class="bf">2(c)</b> Give the periodic solution. Use letters m, b, k, B, w. For [mathjaxinline]x_ p[/mathjaxinline], you should spell out &#8216;phi' for the phase angle. </p>
<p>
<p style="display:inline">[mathjaxinline]x_ p =[/mathjaxinline]</p>
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</p>
<p>
Use the function arctan() to give a formula for [mathjaxinline]\phi[/mathjaxinline]. </p>
<p>
<p style="display:inline">[mathjaxinline]\phi =[/mathjaxinline]</p>
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<p><b class="bf">Note:</b> The answer checker doesn't ask for which quadrants [mathjaxinline]\phi[/mathjaxinline] might be in, but you need to specify this on your paper. </p>
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Problem 3a
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<p><b class="bf">3(a)</b> Give the differential equation modeling [mathjaxinline]x(t)[/mathjaxinline]. It should be in the usual form with all the [mathjaxinline]x[/mathjaxinline] terms on the left (put in first box) and everything else on the right (put in second box). </p>
<p>
For derivatives write [mathjaxinline]x'[/mathjaxinline], [mathjaxinline]x' '[/mathjaxinline], [mathjaxinline]y'[/mathjaxinline] etc. Use the parameters, [mathjaxinline]m[/mathjaxinline], [mathjaxinline]b[/mathjaxinline], [mathjaxinline]k[/mathjaxinline]. You can use [mathjaxinline]g[/mathjaxinline] or 9.8 for the acceleration due to gravity. </p>
<p>
<p style="display:inline">DE: </p>
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<p style="display:inline"> &#8195;&#8195;&#8195;&#8195;[mathjaxinline]=[/mathjaxinline] </p>
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Problem 3b
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<p><b class="bf">3(b)</b> Give the solution as a function of [mathjaxinline]t[/mathjaxinline]. Use the numerical values given and use 9.8 for the acceleration due to gravity. </p>
<p>
Give any decimal numbers you use correct to at least 4 significant places. </p>
<p>
<p style="display:inline">[mathjaxinline]x(t) =[/mathjaxinline] </p>
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Problem 4
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<b class="bf">4(a)</b>
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<h3 class="hd hd-3 problem-header" id="ps3-5a-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-5a-check-problem-progress" tabindex="-1">
Problem 5a
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-5a-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5(a)</b> Enter a comma separated list of roots. Enter each number rounded to <em>exactly</em> 3 decimal places (e.g. 0.01234 should be entered as 0.012). This will be the same values you use in your answer to part (c). Enter complex numbers as, e.g. 2.123 - 3.432*i </p>
<p>
<p style="display:inline">Roots: </p>
<div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps3-5a-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
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</span>
<p id="answer_ps3-5a-check_2_1" class="answer"/>
<div id="display_ps3-5a-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-5a-check_2_1_dynamath" name="input_ps3-5a-check_2_1_dynamath"/>
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data-problem-score="0"
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<h3 class="hd hd-3 problem-header" id="ps3-5b-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-5b-check-problem-progress" tabindex="-1">
Problem 5b
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-5b-check-problem-progress"></div>
<div class="problem">
<div>
<b class="bf">5(b)</b>
<em>No answer checker given.</em>
</div>
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<input type="hidden" name="problem_id" value="Problem 5b" />
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<h3 class="hd hd-3 problem-header" id="ps3-5c-check-problem-title" aria-describedby="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-5c-check-problem-progress" tabindex="-1">
Problem 5c
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-5c-check-problem-progress"></div>
<div class="problem">
<div>
<p><b class="bf">5(c)</b> It is tricky to write a checker for a solution like this. So, you will have to enter your answer carefully in the following way. </p>
<ul class="itemize">
<li>
<p>
Write roots rounded to <em>exactly</em> 3 decimal places. </p>
</li>
<li>
<p>
For the homogeneous part of the solution, write your coefficients as C_{1}, C_{2} etc. The curly braces are important. </p>
</li>
<li>
<p>
So that your pieces are in the same order as the checker, order the modal solutions as follows.<br/>First: real roots in increasing order<br/>Second: complex roots in increasing order of their real parts. For each complex pair: put cosine before sine. </p>
</li>
<li>
<p>
Use the coefficients C_{1}, C_{2} etc. in the same order as the modal solutions. </p>
</li>
</ul>
<p>
Give the general real-valued solution to the DE. <div class="inline" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_ps3-5c-check_2_1" class="text-input-dynamath capa_inputtype inline textline">
<div class="text-input-dynamath_data inline" data-preprocessor="MJxPrep"/>
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<p id="answer_ps3-5c-check_2_1" class="answer"/>
<div id="display_ps3-5c-check_2_1" class="equation">`{::}`</div>
<textarea style="display:none" id="input_ps3-5c-check_2_1_dynamath" name="input_ps3-5c-check_2_1_dynamath"/>
</div>
</div></div> </p>
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Problem 6
</h3>
<div class="problem-progress" id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps3-6-check-problem-progress"></div>
<div class="problem">
<div>
<p>
<b class="bf">6(a)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">6(b)</b>
<em>No answer checker given.</em>
</p>
<p>
<b class="bf">6(c)</b>
<em>No answer checker given.</em>
</p>
</div>
<div class="action">
<input type="hidden" name="problem_id" value="Problem 6" />
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