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{"name": "Ugur's copy of Year 3 Find the Laplace transfrom of ode", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Ugur's copy of Year 3 Find the Laplace transfrom of ode", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Solve the differential equation&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\[\\frac{d^2q}{dt^2}+\\var{a}\\frac{dq}{dt}+\\var{b}q(t)=\\var{c}e^{-\\var{d}t}\\quad&nbsp; &nbsp; \\mbox{where}&nbsp; &nbsp;\\quad q(0)=\\var{f}\\quad \\mbox{and}&nbsp; \\quad q'(0)=\\var{g}\\]&nbsp;</p>\n<p style=\"padding-left: 30px;\">using Laplace Transform method.</p>\n<p style=\"padding-left: 30px;\"></p>", "advice": "<p>a) We are given the ODE</p>\n<p>\\[\\frac{d^2q}{dt^2}+\\var{a}\\frac{dq}{dt}+\\var{b}q(t)=\\var{c}e^{-\\var{d}t}\\quad&nbsp; &nbsp; \\mbox{where}&nbsp; &nbsp;\\quad q(0)=\\var{f}\\quad \\mbox{and}&nbsp; \\quad q'(0)=\\var{g}.\\]</p>\n<p></p>\n<p>Using Laplace Transform Table and teh rule for Laplace Transform of derivatives we transfor the ODE into:&nbsp;</p>\n<p>\\[s^2Q(s)-sq(0)-q'(0)+\\var{a}(s(Q(s)-q(0))+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}.\\]</p>\n<p></p>\n<p>Pluging in initial values we get :</p>\n<p>\\[s^2Q(s)-\\var{f}s-\\var{g}+\\var{a}sQ(s)-\\simplify{{a}*{f}}+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}.\\]</p>\n<p></p>\n<p>Re-arrange to get:</p>\n<p>\\[s^2Q(s)+\\var{a}sQ(s)+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}+\\var{f}s+\\simplify{{g}+{a}*{f}}.\\]</p>\n<p></p>\n<p>Take out&nbsp; $Q(s)$ as a common factor on the left-hand side:</p>\n<p>\\[(s^2+\\var{a}s+\\var{b})Q(s)=\\frac{\\var{c}+(\\var{f}s+\\simplify{{g}+{a}*{f}})(s+\\var{d})}{s+\\var{d}}.\\]</p>\n<p></p>\n<p>Make $Q(s)$ the subject:</p>\n<p>\\[Q(s)=\\frac{\\simplify{{f}s^2+({a}*{f}+{g}+{d}*{f})s+(({g}+{f}*{a})*{d}+{c})}}{(s+\\var{d})(s^2+\\var{a}s+\\var{b})}.\\]</p>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(3 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(10 .. 25#1)", "description": "", "templateType": "randrange", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 10#1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(3 .. 12#1)", "description": "", "templateType": "randrange", "can_override": false}, "g": {"name": "g", "group": "Ungrouped variables", "definition": "random(3 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "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>Find the Laplace transform of the following differential equation and express it \\(Q(s)\\) as a single fraction:</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{d^2q}{dt^2}+\\var{a}\\frac{dq}{dt}+\\var{b}q(t)=\\var{c}e^{-\\var{d}t}\\)&nbsp; &nbsp; where&nbsp; &nbsp;\\(q(0)=\\var{f}\\) and&nbsp; \\(q'(0)=\\var{g}\\)</p>\n<p></p>\n<p>\\(Q(s)=\\)&nbsp;[[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": "({f}s^2+({f}*{a}+{g}+{d}*{f})s+(({g}+{a}*{f})*{d}+{c}))/((s+{d})(s^2+{a}s}+{b}))", "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": "information", "useCustomName": true, "customName": "Part b)", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>OPTIONAL</strong> (This could be tricky!):</p>\n<p>Find the inverse transform of $Q(s)$.</p>"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Ugur Efem", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18261/"}]}