// Numbas version: exam_results_page_options
{"name": "Dampened Harmonic Oscillator  via Laplace Transforms", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Dampened Harmonic Oscillator  via Laplace Transforms", "tags": [], "metadata": {"description": "<h1 class=\"name-header\"><br /><button type=\"button\" class=\"btn btn-link\" title=\"Change this question's name\" data-bind=\"click: mainTabber.setTab('settings')\"><span class=\"glyphicon glyphicon-pencil\"></span></button></h1>\n<div class=\"stamp-header\">\n<div class=\"btn-group\"><button type=\"button\" class=\"stamp btn btn-lg btn-default dropdown-toggle\" data-toggle=\"dropdown\" aria-haspopup=\"true\" aria-expanded=\"false\" data-bind=\"attr: {'data-stamp':current_stamp().status}\" data-stamp=\"ok\">Status:&nbsp;<span data-bind=\"text: current_stamp().status_display\">Ready to use</span>&nbsp;<span class=\"caret\"></span></button></div>\n</div>\n<div class=\"save-status alert alert-success\" data-bind=\"class: autoSave.status_info().class\"></div>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Solve the differential equation&nbsp;<br/>\\[x^{\\prime\\prime} = -\\simplify{{k}*{k}}x - \\var{b}x^\\prime \\quad \\mbox{ with the initial conditions } \\quad x(0) = \\var{l} \\, \\mbox{ and }\\, x^\\prime(0) = 0\\]</p>", "advice": "<p>By re-arranging the given ODE we get:</p>\n<p>\\[x^{\\prime\\prime} +\\var{b}x^\\prime + \\simplify{{k}*{k}}x = 0. \\]</p>\n<p>Then, let $X=\\mathcal{L}\\{x\\}$, and apply Laplace Transform to get&nbsp;</p>\n<p>\\[(s^2X-sx(0)-x^\\prime(0)) +\\var{b}(sX-x(0)) +\\simplify{{k}*{k}}X=0.\\]</p>\n<p>Then plugging in the initial values:</p>\n<p>\\[(s^2X-\\var{l}s) +\\var{b}(sX-\\var{l}) +\\simplify{{k}*{k}}X=0.\\]&nbsp;</p>\n<p>By re-arranging, we get&nbsp;</p>\n<p>\\[X=\\frac{\\var{l}s + \\simplify{2*{k}{l}}}{s^2+\\var{b}s +\\simplify{{k}*{k}}} = \\frac{\\var{l}(s +\\var{k}) + \\simplify{{l}*{k}}}{(s+\\var{k})^2} = \\frac{\\var{l}}{s+\\var{k}} + \\frac{\\simplify{{l}*{k}}}{(s+\\var{k})^2}.\\]</p>\n<p>Finally, by the transform table and the property $\\mathcal{L}\\left\\{t^n g(t)\\right\\} = (-1)^n\\frac{d^nG}{ds^n}$ (where $\\mathcal L\\{g(t)\\} = G(s)$ (Note that one can use the First Shift Theorem instead of this property here).&nbsp;</p>\n<p>\\[x(t) =\\var{l}e^{-4t} + \\simplify{{l}*{k}}te^{-4t} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "2k", "description": "", "templateType": "anything", "can_override": false}, "l": {"name": "l", "group": "Ungrouped variables", "definition": "random(0 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["k", "b", "l"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Take the Laplace transform of the given ODE and write down the transformed equation:&nbsp;</p>\n<p>[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Part a)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "s^2X-{l}s +{b}(sX-{l}) +{{k}*{k}}X=0", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "s", "value": ""}, {"name": "sx", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Re-arrange the transfomed equation to isolate $X$.</p>\n<p>$X=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({l}s + {2*{k}*{l}})/(s^2+{b}s +{{k}*{k}})", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "s", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the partial fractions decomposition of the rational function you found above.</p>\n<p>$X=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({l})/(s+{k}) + ({{l}*{k}})/((s+{k})^2)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "s", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the inverse transform of $X(s)$:&nbsp;</p>\n<p>$x(t)=$[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{l}e^(-4t) + {{l}*{k}}te^(-4t)", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "t", "value": ""}, {"name": "te", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}