<div class="xblock xblock-public_view xblock-public_view-vertical" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="False" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@vertical+block@ps8-checkvertical" data-init="VerticalStudentView" data-graded="False" data-runtime-version="1" data-block-type="vertical" data-request-token="709aff68383a11f0982f12f8c5a42579"> <h2 class="hd hd-2 unit-title">Pset 8 checker</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+ES.1803+2023_Fall+type@html+block@618181e2b14f40a099bb36ab5bd59b5a"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="False" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@html+block@618181e2b14f40a099bb36ab5bd59b5a" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="html" data-request-token="709aff68383a11f0982f12f8c5a42579"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p><a href="/assets/courseware/v1/57821e07416a58dc5674ac0c06c5432b/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/mites_1803_s24_ps8.pdf" target="[object Object]">Problem set 8 (PDF)</a></p> <p></p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-1-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-1-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="709aff68383a11f0982f12f8c5a42579"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps8-1-check" class="problems-wrapper" role="group" aria-labelledby="ps8-1-check-problem-title" data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-1-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-1-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps8-1-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-1-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 1 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-1-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; To check your answers put them in the appropriate box and click the &amp;#8216;Check&#39; button. The checker can do arithmetic. So, 3^2, [mathjaxinline]5/3[/mathjaxinline] and [mathjaxinline]3*4[/mathjaxinline] are okay. &lt;/p&gt; &lt;p&gt; Unless otherwise stated the checker expects numerical answers to 3 significant digits. &lt;/p&gt; &lt;p&gt; For formulas you should use Matlab notation. Remember to use a * for multiplication, i.e. [mathjaxinline]3*x[/mathjaxinline] &lt;b class=&#34;bf&#34;&gt;NOT &lt;/b&gt; [mathjaxinline]3x[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; The checker knows many of the standard functions. Use exp(x) and ln(x) for the exponential and natural log functions. &lt;/p&gt; &lt;p&gt; The formula checker expects a specific letter for the variables, so be sure to use the one specified. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1(a)&lt;/b&gt; On your paper, your solution should be given in the usual form with coefficients [mathjaxinline]c_1[/mathjaxinline] and [mathjaxinline]c_2[/mathjaxinline] and vectors which are functions of [mathjaxinline]t[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; For the problem set checker, we will check your eigenvalues and eigenvectors. Give two eigenvalue/eigenvector pairs. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;Warning:&lt;/b&gt; The checker is not that clever. It will mark it correct if you give the same pair twice. The graders will undoubtedly mark this wrong. You should have two distinct pairs. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;Give one pair.&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;multi-inputs-group&#34; role=&#34;group&#34;&gt;&lt;p style=&#34;display:inline&#34;&gt;Eigenvalue: &lt;div id=&#34;inputtype_ps8-1-check_2_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_2_1&#34; id=&#34;input_ps8-1-check_2_1&#34; aria-describedby=&#34;status_ps8-1-check_2_1&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&gt; &lt;br/&gt; &lt;p style=&#34;display:inline&#34;&gt;Eigenvector: &lt;div id=&#34;inputtype_ps8-1-check_2_2&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_2_2&#34; id=&#34;input_ps8-1-check_2_2&#34; aria-describedby=&#34;status_ps8-1-check_2_2&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_2_2&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_2_2&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_2_2&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;Give a second pair.&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;multi-inputs-group&#34; role=&#34;group&#34;&gt;&lt;p style=&#34;display:inline&#34;&gt;Eigenvalue: &lt;div id=&#34;inputtype_ps8-1-check_3_1&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_3_1&#34; id=&#34;input_ps8-1-check_3_1&#34; aria-describedby=&#34;status_ps8-1-check_3_1&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&gt; &lt;br/&gt; &lt;p style=&#34;display:inline&#34;&gt;Eigenvector: &lt;div id=&#34;inputtype_ps8-1-check_3_2&#34; class=&#34; capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_3_2&#34; id=&#34;input_ps8-1-check_3_2&#34; aria-describedby=&#34;status_ps8-1-check_3_2&#34; value=&#34;&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_3_2&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_3_2&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_3_2&#34; class=&#34;answer&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/p&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;1(b)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;1(c)&lt;/b&gt; We will assume you have correctly found and included the homogeneous solution in your general solutions. For the particular solutions, give [mathjaxinline]x_ p(t)[/mathjaxinline] and [mathjaxinline]y_ p(t)[/mathjaxinline] as the sums indicated. &lt;/p&gt; &lt;p&gt; Use the variables n and phi_n, where phi_n is the phase lag in the nth Fourier term &lt;/p&gt; &lt;p&gt; We give you a nice hint that the sums are a DC term + a sum over [mathjaxinline]n[/mathjaxinline] odd! &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]x_ p(t) =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-1-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_4_1&#34; id=&#34;input_ps8-1-check_4_1&#34; aria-describedby=&#34;status_ps8-1-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;10&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-1-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-1-check_4_1_dynamath&#34; name=&#34;input_ps8-1-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt;+ &lt;big class=&#34;xxlarge&#34;&gt;[mathjaxinline]\sum _{n\text { odd}}[/mathjaxinline]&lt;/big&gt;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-1-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_5_1&#34; id=&#34;input_ps8-1-check_5_1&#34; aria-describedby=&#34;status_ps8-1-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-1-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-1-check_5_1_dynamath&#34; name=&#34;input_ps8-1-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]y_ p(t) =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-1-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_6_1&#34; id=&#34;input_ps8-1-check_6_1&#34; aria-describedby=&#34;status_ps8-1-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;10&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-1-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-1-check_6_1_dynamath&#34; name=&#34;input_ps8-1-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;p style=&#34;display:inline&#34;&gt;+ &lt;big class=&#34;xxlarge&#34;&gt;[mathjaxinline]\sum _{n\text { odd}}[/mathjaxinline]&lt;/big&gt;&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-1-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_7_1&#34; id=&#34;input_ps8-1-check_7_1&#34; aria-describedby=&#34;status_ps8-1-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-1-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-1-check_7_1_dynamath&#34; name=&#34;input_ps8-1-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the phase lag for the nth Fourier term. Use the variable n. You can use the function arctan2(x,y) for the inverse tan function. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\phi _ n =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 7&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-1-check_8_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-1-check_8_1&#34; id=&#34;input_ps8-1-check_8_1&#34; aria-describedby=&#34;status_ps8-1-check_8_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-1-check_8_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-1-check_8_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-1-check_8_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-1-check_8_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-1-check_8_1_dynamath&#34; name=&#34;input_ps8-1-check_8_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;1(d)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 1&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps8-1-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps8-1-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-1-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-1-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-1-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-2-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-2-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="709aff68383a11f0982f12f8c5a42579"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps8-2-check" class="problems-wrapper" role="group" aria-labelledby="ps8-2-check-problem-title" data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-2-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-2-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps8-2-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-2-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 2 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-2-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 2&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps8-2-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps8-2-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-2-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-2-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-2-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-3" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-3-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-3-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="709aff68383a11f0982f12f8c5a42579"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps8-3-check" class="problems-wrapper" role="group" aria-labelledby="ps8-3-check-problem-title" data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-3-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-3-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps8-3-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-3-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 3 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-3-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;3(a)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;3(b)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;3(c)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;3(d)&lt;/b&gt; There is no checker for this problem either, but we can ask a question about superposition of boundary conditions that might help guide you. &lt;/p&gt; &lt;p&gt; Suppose [mathjaxinline]u_1(0,t) = 2[/mathjaxinline] and [mathjaxinline]u_2(0,t) = 5[/mathjaxinline]. Let [mathjaxinline]u = u_1 + u_2[/mathjaxinline]. What does [mathjaxinline]u(0,t)[/mathjaxinline] equal? &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]u(0,t) =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-3-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-3-check_2_1&#34; id=&#34;input_ps8-3-check_2_1&#34; aria-describedby=&#34;status_ps8-3-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;15&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-3-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-3-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-3-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-3-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-3-check_2_1_dynamath&#34; name=&#34;input_ps8-3-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 3&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps8-3-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps8-3-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-3-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-3-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-3-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-4" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-4-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-4-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="709aff68383a11f0982f12f8c5a42579"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps8-4-check" class="problems-wrapper" role="group" aria-labelledby="ps8-4-check-problem-title" data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-4-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-4-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps8-4-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-4-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 4 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-4-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;4(a)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;4(b)&lt;/b&gt; Give the steady-state solution which satisfies both the inhomogeneous PDE and the inhomogeneous boundary condition. Use the variables x and T_0. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]u_ p(x,t)[/mathjaxinline] = &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-4-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-4-check_2_1&#34; id=&#34;input_ps8-4-check_2_1&#34; aria-describedby=&#34;status_ps8-4-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-4-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-4-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-4-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-4-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-4-check_2_1_dynamath&#34; name=&#34;input_ps8-4-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;4(c)&lt;/b&gt; Give the first 4 &lt;b class=&#34;bf&#34;&gt;non-zero&lt;/b&gt; terms in the series for the solution, [mathjaxinline]u_ h(x,t)[/mathjaxinline], to the homogeneous equation. Your indexing of the coefficients might be different, we will use the coefficients [mathjaxinline]c_1[/mathjaxinline], [mathjaxinline]c_2[/mathjaxinline], [mathjaxinline]c_3[/mathjaxinline], [mathjaxinline]c_4[/mathjaxinline] for these terms. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]u_ h(x,t) = c_1[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-4-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-4-check_3_1&#34; id=&#34;input_ps8-4-check_3_1&#34; aria-describedby=&#34;status_ps8-4-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-4-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-4-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-4-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-4-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-4-check_3_1_dynamath&#34; name=&#34;input_ps8-4-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\qquad + c_2[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-4-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-4-check_4_1&#34; id=&#34;input_ps8-4-check_4_1&#34; aria-describedby=&#34;status_ps8-4-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-4-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-4-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-4-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-4-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-4-check_4_1_dynamath&#34; name=&#34;input_ps8-4-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\qquad + c_3[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 4&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-4-check_5_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-4-check_5_1&#34; id=&#34;input_ps8-4-check_5_1&#34; aria-describedby=&#34;status_ps8-4-check_5_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-4-check_5_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-4-check_5_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-4-check_5_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-4-check_5_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-4-check_5_1_dynamath&#34; name=&#34;input_ps8-4-check_5_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\qquad + c_4[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 5&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-4-check_6_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-4-check_6_1&#34; id=&#34;input_ps8-4-check_6_1&#34; aria-describedby=&#34;status_ps8-4-check_6_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;50&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-4-check_6_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-4-check_6_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-4-check_6_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-4-check_6_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-4-check_6_1_dynamath&#34; name=&#34;input_ps8-4-check_6_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;4(d)&lt;/b&gt; Give the general solution [mathjaxinline]u[/mathjaxinline] in terms of the particular solution u_p and homogeneous solution u_h. (This is just as easy as it seems.) &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]u =[/mathjaxinline] &lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 6&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-4-check_7_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-4-check_7_1&#34; id=&#34;input_ps8-4-check_7_1&#34; aria-describedby=&#34;status_ps8-4-check_7_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-4-check_7_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-4-check_7_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-4-check_7_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-4-check_7_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-4-check_7_1_dynamath&#34; name=&#34;input_ps8-4-check_7_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;4(e)&lt;/b&gt; &lt;b class=&#34;bf&#34;&gt;(Extra credit) No checker given.&lt;/b&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 4&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps8-4-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps8-4-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-4-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-4-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-4-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> <div class="vert vert-5" data-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-5-check"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-runtime-class="LmsRuntime" data-course-id="course-v1:MITx+ES.1803+2023_Fall" data-has-score="True" data-usage-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-5-check" data-init="XBlockToXModuleShim" data-graded="False" data-runtime-version="1" data-block-type="problem" data-request-token="709aff68383a11f0982f12f8c5a42579"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_ps8-5-check" class="problems-wrapper" role="group" aria-labelledby="ps8-5-check-problem-title" data-problem-id="block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-5-check" data-url="/courses/course-v1:MITx+ES.1803+2023_Fall/xblock/block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-5-check/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;ps8-5-check-problem-title&#34; aria-describedby=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-5-check-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Problem 5 &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:MITx+ES.1803+2023_Fall+type@problem+block@ps8-5-check-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Don&#39;t forget that in this problem we have [mathjaxinline]L=1[/mathjaxinline], [mathjaxinline]a=2[/mathjaxinline], [mathjaxinline]b=7[/mathjaxinline], [mathjaxinline]g(x) = 0[/mathjaxinline]. &lt;/p&gt; &lt;p&gt;&lt;b class=&#34;bf&#34;&gt;5(a)&lt;/b&gt; This is a tricky problem to write a checker for, let&#39;s try the following. Suppose the initial condition, [mathjaxinline]f(x)[/mathjaxinline], has sine series &lt;/p&gt; &lt;table id=&#34;a0000000002&#34; class=&#34;equation&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; cellpadding=&#34;7&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr&gt; &lt;td class=&#34;equation&#34; style=&#34;width:80%; border:none&#34;&gt;[mathjax]f(x) = \sum _{n=1}^\infty c_ n\sin (n\pi x) \quad \text { on $0\leq x\leq 1$}.[/mathjax]&lt;/td&gt; &lt;td class=&#34;eqnnum&#34; style=&#34;width:20%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Give the Fourier solution to the wave equation with the given BC and IC. &lt;/p&gt; &lt;p&gt; Use c_n for the sine coefficients of [mathjaxinline]f[/mathjaxinline] and also use the variables x, t, n. &lt;/p&gt; &lt;p&gt; One more hint to make this easier to enter: The terms in your answer should have terms [mathjaxinline]\cos (\beta _ n t)[/mathjaxinline] and [mathjaxinline]\sin (\beta _ n t)[/mathjaxinline], where [mathjaxinline]\beta _ n[/mathjaxinline] is a function of [mathjaxinline]n[/mathjaxinline]. In your answer you can write beta(n) instead of typing the formula several times. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\displaystyle y(x,t) = \sum _{n=1}^{\infty }[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 1&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-5-check_2_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-5-check_2_1&#34; id=&#34;input_ps8-5-check_2_1&#34; aria-describedby=&#34;status_ps8-5-check_2_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;70&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-5-check_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-5-check_2_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-5-check_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-5-check_2_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-5-check_2_1_dynamath&#34; name=&#34;input_ps8-5-check_2_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Give the formula for [mathjaxinline]\beta _ n[/mathjaxinline]. (Use the function abs() for absolute value.) &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\beta _ n =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-5-check_3_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-5-check_3_1&#34; id=&#34;input_ps8-5-check_3_1&#34; aria-describedby=&#34;status_ps8-5-check_3_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-5-check_3_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-5-check_3_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-5-check_3_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-5-check_3_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-5-check_3_1_dynamath&#34; name=&#34;input_ps8-5-check_3_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; On your paper you will need to give the integral formula for [mathjaxinline]c_ n[/mathjaxinline]. Here, give the data for the integration as a vector in the form [lower_limit, upper_limit, integrand]. Use x as the variable of integration and f(x) for the arbitrary function in the initial condition [mathjaxinline]y(x,0) = f(x)[/mathjaxinline]. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]c_ n =[/mathjaxinline]&lt;/p&gt; &lt;div class=&#34;inline&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 3&#34; role=&#34;group&#34;&gt;&lt;div id=&#34;inputtype_ps8-5-check_4_1&#34; class=&#34;text-input-dynamath capa_inputtype inline textline&#34;&gt; &lt;div class=&#34;text-input-dynamath_data inline&#34; data-preprocessor=&#34;MJxPrep&#34;/&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/assets/courseware/v1/12474df7814ecb7a9a4088614ed78d4c/asset-v1:MITx+ES.1803+2023_Fall+type@asset+block/MJxPrep.js&#34;/&gt; &lt;div class=&#34;unanswered inline&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_ps8-5-check_4_1&#34; id=&#34;input_ps8-5-check_4_1&#34; aria-describedby=&#34;status_ps8-5-check_4_1&#34; value=&#34;&#34; class=&#34;math&#34; size=&#34;60&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_ps8-5-check_4_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_ps8-5-check_4_1&#34; data-tooltip=&#34;Not yet answered.&#34;&gt; &lt;span class=&#34;sr&#34;&gt;unanswered&lt;/span&gt;&lt;span class=&#34;status-icon&#34; aria-hidden=&#34;true&#34;/&gt; &lt;/span&gt; &lt;p id=&#34;answer_ps8-5-check_4_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;display_ps8-5-check_4_1&#34; class=&#34;equation&#34;&gt;`{::}`&lt;/div&gt; &lt;textarea style=&#34;display:none&#34; id=&#34;input_ps8-5-check_4_1_dynamath&#34; name=&#34;input_ps8-5-check_4_1_dynamath&#34;/&gt; &lt;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;b class=&#34;bf&#34;&gt;5(b)&lt;/b&gt; &lt;em&gt;No answer checker given.&lt;/em&gt; &lt;/p&gt; &lt;/div&gt; &lt;div class=&#34;action&#34;&gt; &lt;input type=&#34;hidden&#34; name=&#34;problem_id&#34; value=&#34;Problem 5&#34; /&gt; &lt;div class=&#34;submit-attempt-container&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;submit btn-brand&#34; data-submitting=&#34;Submitting&#34; data-value=&#34;Submit&#34; data-should-enable-submit-button=&#34;True&#34; aria-describedby=&#34;submission_feedback_ps8-5-check&#34; &gt; &lt;span class=&#34;submit-label&#34;&gt;Submit&lt;/span&gt; &lt;/button&gt; &lt;div class=&#34;submission-feedback&#34; id=&#34;submission_feedback_ps8-5-check&#34;&gt; &lt;span class=&#34;sr&#34;&gt;Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.&lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;problem-action-buttons-wrapper&#34;&gt; &lt;span class=&#34;problem-action-button-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;save problem-action-btn btn-default btn-small&#34; data-value=&#34;Save&#34;&gt; &lt;span class=&#34;icon fa fa-floppy-o&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span aria-hidden=&#34;true&#34;&gt;Save&lt;/span&gt; &lt;span class=&#34;sr&#34;&gt;Save your answer&lt;/span&gt; &lt;/button&gt; &lt;/span&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-gentle-alert is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-exclamation-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-5-check-problem-title&#34;&gt; &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification warning notification-save is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-save&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-5-check-problem-title&#34;&gt;None &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;notification general notification-show-answer is-hidden&#34; tabindex=&#34;-1&#34;&gt; &lt;span class=&#34;icon fa fa-info-circle&#34; aria-hidden=&#34;true&#34;&gt;&lt;/span&gt; &lt;span class=&#34;notification-message&#34; aria-describedby=&#34;ps8-5-check-problem-title&#34;&gt;Answers are displayed within the problem &lt;/span&gt; &lt;div class=&#34;notification-btn-wrapper&#34;&gt; &lt;button type=&#34;button&#34; class=&#34;btn btn-default btn-small notification-btn review-btn sr&#34;&gt;Review&lt;/button&gt; &lt;/div&gt; &lt;/div&gt; &lt;/div&gt; " data-graded="False"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading&hellip;</span> </p> </div> </div> </div> </div> </div>