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{"name": "Find the Laplace transform of ode", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find the Laplace transform of ode", "variables": {"d": {"definition": "random(3..8#1)", "name": "d", "group": "Ungrouped variables", "templateType": "randrange", "description": ""}, "b": {"definition": "random(10..25#1)", "name": "b", "group": "Ungrouped variables", "templateType": "randrange", "description": ""}, "g": {"definition": "random(3..9#1)", "name": "g", "group": "Ungrouped variables", "templateType": "randrange", "description": ""}, "a": {"definition": "random(2..10#1)", "name": "a", "group": "Ungrouped variables", "templateType": "randrange", "description": ""}, "f": {"definition": "random(1..6#1)", "name": "f", "group": "Ungrouped variables", "templateType": "randrange", "description": ""}, "c": {"definition": "random(3..12#1)", "name": "c", "group": "Ungrouped variables", "templateType": "randrange", "description": ""}}, "advice": "<p>\\(\\frac{d^2x}{dt^2}+\\var{a}\\frac{dx}{dt}+\\var{b}x(t)=\\var{c}e^{-\\var{d}t}\\)&nbsp; &nbsp; where&nbsp; &nbsp;\\(x(0)=\\var{f}\\) and&nbsp; \\(x'(0)=\\var{g}\\)</p>\n<p></p>\n<p>\\(s^2X(s)-sx(0)-x'(0)+\\var{a}(s(X(s)-x(0))+\\var{b}X(s)=\\frac{\\var{c}}{s+\\var{d}}\\)</p>\n<p></p>\n<p>\\(s^2X(s)-\\var{f}s-\\var{g}+\\var{a}sX(s)-\\var{a}*\\var{f}+\\var{b}X(s)=\\frac{\\var{c}}{s+\\var{d}}\\)</p>\n<p></p>\n<p>\\(s^2X(s)+\\var{a}sX(s)+\\var{b}X(s)=\\frac{\\var{c}}{s+\\var{d}}+\\var{f}s+\\simplify{{g}+{a}*{f}}\\)</p>\n<p></p>\n<p>\\((s^2+\\var{a}s+\\var{b})X(s)=\\frac{\\var{c}+(\\var{f}s+\\simplify{{g}+{a}*{f}})(s+\\var{d})}{s+\\var{d}}\\)</p>\n<p></p>\n<p>\\(X(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>", "variable_groups": [], "statement": "<p>Find the Laplace transform of the following differential equation and express it as a single fraction:</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{d^2x}{dt^2}+\\var{a}\\frac{dx}{dt}+\\var{b}x(t)=\\var{c}e^{-\\var{d}t}\\)&nbsp; &nbsp; where&nbsp; &nbsp;\\(x(0)=\\var{f}\\) and&nbsp; \\(x'(0)=\\var{g}\\)</p>", "functions": {}, "preamble": {"css": "", "js": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "parts": [{"scripts": {}, "type": "gapfill", "sortAnswers": false, "extendBaseMarkingAlgorithm": true, "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "scripts": {}, "type": "jme", "failureRate": 1, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "checkVariableNames": false, "unitTests": [], "showFeedbackIcon": true, "variableReplacements": [], "vsetRange": [0, 1], "showPreview": true, "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "marks": "3", "vsetRangePoints": 5, "answer": "({f}s^2+({f}*{a}+{g}+{d}*{f})s+(({g}+{a}*{f})*{d}+{c}))/((s+{d})(s^2+{a}s}+{b}))"}], "unitTests": [], "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "prompt": "<p>\\(X(s)=\\)&nbsp;[[0]]</p>"}], "rulesets": {}, "extensions": [], "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Marie Nicholson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1799/"}]}