<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-request-token="e198877eea7b11efbdc412d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:OCW+18.031+2019_Spring" data-init="VerticalStudentView" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@rec3-tab1"> <h2 class="hd hd-2 unit-title">3.1. Laplace transform of a damped sinusoid.</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab1-problem1"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="False" data-request-token="e198877eea7b11efbdc412d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:OCW+18.031+2019_Spring" data-init="XBlockToXModuleShim" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab1-problem1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_rec3-tab1-problem1" class="problems-wrapper" role="group" aria-labelledby="rec3-tab1-problem1-problem-title" data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab1-problem1" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab1-problem1/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="10.0" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;rec3-tab1-problem1-problem-title&#34; aria-describedby=&#34;block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab1-problem1-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Activity &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab1-problem1-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; (We encourage discussing this problem on the forum. It is worth more points than a regular problem.) &lt;/p&gt; &lt;p&gt; Damped sinusoids are important signals! Compute [mathjaxinline]\mathcal{L}(e^{at}\cos (\omega t);s)[/mathjaxinline] where [mathjaxinline]a[/mathjaxinline] is real by the method used on the previous page. Don&#39;t forget to find the region of convergence. &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{L}(e^{at}\cos (\omega t);s)=[/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;formulaequationinput_rec3-tab1-problem1_2_1&#34; class=&#34;inputtype formulaequationinput&#34; style=&#34;display:inline-block;vertical-align:top&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_rec3-tab1-problem1_2_1&#34; id=&#34;input_rec3-tab1-problem1_2_1&#34; data-input-id=&#34;rec3-tab1-problem1_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_rec3-tab1-problem1_2_1&#34; size=&#34;20&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_rec3-tab1-problem1_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_rec3-tab1-problem1_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_rec3-tab1-problem1_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_rec3-tab1-problem1_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; Region of convergence: &lt;div class=&#34;wrapper-problem-response&#34; tabindex=&#34;-1&#34; aria-label=&#34;Question 2&#34; role=&#34;group&#34;&gt;&lt;div class=&#34;choicegroup capa_inputtype&#34; id=&#34;inputtype_rec3-tab1-problem1_3_1&#34;&gt; &lt;fieldset aria-describedby=&#34;status_rec3-tab1-problem1_3_1&#34;&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_rec3-tab1-problem1_3_1&#34; id=&#34;input_rec3-tab1-problem1_3_1_choice_1&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_1&#34;/&gt;&lt;label id=&#34;rec3-tab1-problem1_3_1-choice_1-label&#34; for=&#34;input_rec3-tab1-problem1_3_1_choice_1&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_rec3-tab1-problem1_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]|s|&amp;gt;a[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_rec3-tab1-problem1_3_1&#34; id=&#34;input_rec3-tab1-problem1_3_1_choice_2&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_2&#34;/&gt;&lt;label id=&#34;rec3-tab1-problem1_3_1-choice_2-label&#34; for=&#34;input_rec3-tab1-problem1_3_1_choice_2&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_rec3-tab1-problem1_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]|s|&amp;gt;0[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_rec3-tab1-problem1_3_1&#34; id=&#34;input_rec3-tab1-problem1_3_1_choice_3&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_3&#34;/&gt;&lt;label id=&#34;rec3-tab1-problem1_3_1-choice_3-label&#34; for=&#34;input_rec3-tab1-problem1_3_1_choice_3&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_rec3-tab1-problem1_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]\mathrm{Re\, }s &amp;gt; a[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_rec3-tab1-problem1_3_1&#34; id=&#34;input_rec3-tab1-problem1_3_1_choice_4&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_4&#34;/&gt;&lt;label id=&#34;rec3-tab1-problem1_3_1-choice_4-label&#34; for=&#34;input_rec3-tab1-problem1_3_1_choice_4&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_rec3-tab1-problem1_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]\mathrm{Re\, }s &amp;gt; 0[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;div class=&#34;field&#34;&gt; &lt;input type=&#34;radio&#34; name=&#34;input_rec3-tab1-problem1_3_1&#34; id=&#34;input_rec3-tab1-problem1_3_1_choice_5&#34; class=&#34;field-input input-radio&#34; value=&#34;choice_5&#34;/&gt;&lt;label id=&#34;rec3-tab1-problem1_3_1-choice_5-label&#34; for=&#34;input_rec3-tab1-problem1_3_1_choice_5&#34; class=&#34;response-label field-label label-inline&#34; aria-describedby=&#34;status_rec3-tab1-problem1_3_1&#34;&gt; &lt;text&gt; [mathjaxinline]\mathrm{Im\, }s &amp;gt; 0[/mathjaxinline]&lt;/text&gt; &lt;/label&gt; &lt;/div&gt; &lt;span id=&#34;answer_rec3-tab1-problem1_3_1&#34;/&gt; &lt;/fieldset&gt; &lt;div class=&#34;indicator-container&#34;&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_rec3-tab1-problem1_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;/div&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_rec3-tab1-problem1_solution_1&#34;/&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;Activity&#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_rec3-tab1-problem1&#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_rec3-tab1-problem1&#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;rec3-tab1-problem1-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;rec3-tab1-problem1-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;rec3-tab1-problem1-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>

<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-request-token="e198877eea7b11efbdc412d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:OCW+18.031+2019_Spring" data-init="VerticalStudentView" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@vertical+block@rec3-tab2"> <h2 class="hd hd-2 unit-title">3.2. Laplace transform of higher derivatives.</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:OCW+18.031+2019_Spring+type@html+block@rec3-tab2-text1"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="False" data-request-token="e198877eea7b11efbdc412d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:OCW+18.031+2019_Spring" data-init="XBlockToXModuleShim" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@html+block@rec3-tab2-text1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <p> We can go on to compute the effect of Laplace transform on the second derivative. To make the calculation clearer, let's write [mathjaxinline]g(t)=f'(t)[/mathjaxinline] and [mathjaxinline]G(s)[/mathjaxinline] for its Laplace transform; so we know that [mathjaxinline]G(s)=sF(s)-f(0)[/mathjaxinline]. Then </p><table id="a0000000336" cellpadding="7" width="100%" cellspacing="0" class="eqnarray" style="table-layout:auto"><tr id="a0000000337"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none"> [mathjaxinline]\displaystyle f^{\prime \prime }(t)=g'(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:left; border:none"> [mathjaxinline]\displaystyle \, sG(s)-g(0)[/mathjaxinline] </td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000338"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">   </td><td style="vertical-align:middle; text-align:center; border:none"> [mathjaxinline]\displaystyle =[/mathjaxinline] </td><td style="vertical-align:middle; text-align:left; border:none"> [mathjaxinline]\displaystyle \, s(sF(s)-f(0))-g(0)[/mathjaxinline] </td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr><tr id="a0000000339"><td style="width:40%; border:none"> </td><td style="vertical-align:middle; text-align:right; border:none">   </td><td style="vertical-align:middle; text-align:center; border:none"> [mathjaxinline]\displaystyle =[/mathjaxinline] </td><td style="vertical-align:middle; text-align:left; border:none"> [mathjaxinline]\displaystyle \, s^2F(s)-f(0)s-f'(0)\, .[/mathjaxinline] </td><td style="width:40%; border:none"> </td><td style="width:20%; border:none" class="eqnnum"> </td></tr></table><p> This process continues; we find that [mathjaxinline]\mathcal{L}(f^{(n)}(t);s)[/mathjaxinline] depends upon [mathjaxinline]f(0),f'(0),\ldots ,f^{(n-1)}(0)[/mathjaxinline]: </p><table id="a0000000340" class="equation" width="100%" cellspacing="0" cellpadding="7" style="table-layout:auto"><tr><td class="equation" style="width:80%; border:none">[mathjax]f^{(n)}(t)\rightsquigarrow s^ nF(s)-\left(f(0)s^{n-1}+f'(0)s^{n-2}+\cdots +f^{(n-1)}(0)\right)\, .[/mathjax]</td><td class="eqnnum" style="width:20%; border:none"> </td></tr></table> </div> </div> <div class="vert vert-1" data-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab2-problem1"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="False" data-request-token="e198877eea7b11efbdc412d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:OCW+18.031+2019_Spring" data-init="XBlockToXModuleShim" data-usage-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab2-problem1"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_rec3-tab2-problem1" class="problems-wrapper" role="group" aria-labelledby="rec3-tab2-problem1-problem-title" data-problem-id="block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab2-problem1" data-url="/courses/course-v1:OCW+18.031+2019_Spring/xblock/block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab2-problem1/handler/xmodule_handler" data-problem-score="0" data-problem-total-possible="1" data-attempts-used="0" data-content=" &lt;h3 class=&#34;hd hd-3 problem-header&#34; id=&#34;rec3-tab2-problem1-problem-title&#34; aria-describedby=&#34;block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab2-problem1-problem-progress&#34; tabindex=&#34;-1&#34;&gt; Find the Laplace Transform &lt;/h3&gt; &lt;div class=&#34;problem-progress&#34; id=&#34;block-v1:OCW+18.031+2019_Spring+type@problem+block@rec3-tab2-problem1-problem-progress&#34;&gt;&lt;/div&gt; &lt;div class=&#34;problem&#34;&gt; &lt;div&gt; &lt;p&gt; Suppose that the Laplace transform of a function [mathjaxinline]x(t)[/mathjaxinline] is [mathjaxinline]X(s)[/mathjaxinline]. Moreover, suppose that &lt;/p&gt; &lt;table id=&#34;a0000000341&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; class=&#34;eqnarray&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000342&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle x(0)[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle 2[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;width:20%; border:none;text-align:right&#34; class=&#34;eqnnum&#34;&gt;(3.33)&lt;/td&gt; &lt;/tr&gt; &lt;tr id=&#34;a0000000343&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle \dot x(0)[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle -1[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;width:20%; border:none;text-align:right&#34; class=&#34;eqnnum&#34;&gt;(3.34)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; Use linearity and the [mathjaxinline]t[/mathjaxinline]-derivative formula to find the Laplace transform of the function: &lt;/p&gt; &lt;table id=&#34;a0000000344&#34; cellpadding=&#34;7&#34; width=&#34;100%&#34; cellspacing=&#34;0&#34; class=&#34;eqnarray&#34; style=&#34;table-layout:auto&#34;&gt; &lt;tr id=&#34;a0000000345&#34;&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:right; border:none&#34;&gt; [mathjaxinline]\displaystyle y(t)[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:center; border:none&#34;&gt; [mathjaxinline]\displaystyle =[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;vertical-align:middle; text-align:left; border:none&#34;&gt; [mathjaxinline]\displaystyle \ddot x + \dot x +2 x.[/mathjaxinline] &lt;/td&gt; &lt;td style=&#34;width:40%; border:none&#34;&gt;&amp;#160;&lt;/td&gt; &lt;td style=&#34;width:20%; border:none;text-align:right&#34; class=&#34;eqnnum&#34;&gt;(3.35)&lt;/td&gt; &lt;/tr&gt; &lt;/table&gt; &lt;p&gt; (Type &lt;b class=&#34;bf&#34;&gt;X&lt;/b&gt; for [mathjaxinline]X(s)[/mathjaxinline]. Note that the answer is case sensitive.) &lt;/p&gt; &lt;p&gt; &lt;p style=&#34;display:inline&#34;&gt;[mathjaxinline]\mathcal{L}\left(y(t); s\right)=[/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;formulaequationinput_rec3-tab2-problem1_2_1&#34; class=&#34;inputtype formulaequationinput&#34; style=&#34;display:inline-block;vertical-align:top&#34;&gt; &lt;div class=&#34;unanswered&#34;&gt; &lt;input type=&#34;text&#34; name=&#34;input_rec3-tab2-problem1_2_1&#34; id=&#34;input_rec3-tab2-problem1_2_1&#34; data-input-id=&#34;rec3-tab2-problem1_2_1&#34; value=&#34;&#34; aria-describedby=&#34;status_rec3-tab2-problem1_2_1&#34; size=&#34;40&#34;/&gt; &lt;span class=&#34;trailing_text&#34; id=&#34;trailing_text_rec3-tab2-problem1_2_1&#34;/&gt; &lt;span class=&#34;status unanswered&#34; id=&#34;status_rec3-tab2-problem1_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_rec3-tab2-problem1_2_1&#34; class=&#34;answer&#34;/&gt; &lt;div id=&#34;input_rec3-tab2-problem1_2_1_preview&#34; class=&#34;equation&#34;&gt; \(\) &lt;img src=&#34;/static/images/spinner.bc34f953403f.gif&#34; class=&#34;loading&#34; alt=&#34;Loading&#34;/&gt; &lt;/div&gt; &lt;/div&gt; &lt;div class=&#34;script_placeholder&#34; data-src=&#34;/static/js/capa/src/formula_equation_preview.b1967ab28c31.js&#34;/&gt; &lt;/div&gt;&lt;/div&gt; &lt;/p&gt; &lt;p&gt; &lt;div class=&#34;solution-span&#34;&gt; &lt;span id=&#34;solution_rec3-tab2-problem1_solution_1&#34;/&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;Find the Laplace Transform&#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_rec3-tab2-problem1&#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_rec3-tab2-problem1&#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;rec3-tab2-problem1-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;rec3-tab2-problem1-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;rec3-tab2-problem1-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>