// Numbas version: exam_results_page_options
{"name": "Laplace:  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": [{"metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "variables": {"c": {"definition": "random(3..12#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "c"}, "d": {"definition": "random(3..8#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "d"}, "g": {"definition": "random(3..9#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "g"}, "b": {"definition": "random(10..25#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "b"}, "a": {"definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "a"}, "f": {"definition": "random(1..6#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "f"}}, "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>", "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "name": "Laplace:  Find the Laplace transform of ode", "rulesets": {}, "preamble": {"css": "", "js": ""}, "tags": [], "variable_groups": [], "statement": "<p></p>\n<p>Given the&nbsp;differential equation:</p>\n<p></p>\n<p style=\"padding-left: 30px; text-align: center;\">\\(\\dfrac{d^2x}{dt^2}+\\var{a}\\dfrac{dx}{dt}+\\var{b}x=\\var{c}e^{-\\var{d}t}\\)&nbsp; &nbsp; where&nbsp; &nbsp;\\(x(0)=\\var{f}\\) and&nbsp; \\(x'(0)=\\var{g}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<ol>\n<li>Take&nbsp;the<strong>&nbsp;Laplace transform</strong>&nbsp;of the equation,</li>\n<li>Substitute in the <strong>conditions</strong>,</li>\n<li><strong>Rearrange</strong> for&nbsp;\\(\\mathscr{L}(x)\\) and then express the <strong>right-hand side as&nbsp;a single fraction</strong>.</li>\n<li>Enter this fraction in the box below.</li>\n</ol>\n<p></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"type": "gapfill", "marks": 0, "unitTests": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "<p>\\(\\mathscr{L}(x)=\\)&nbsp;[[0]]</p>", "customMarkingAlgorithm": "", "gaps": [{"type": "jme", "marks": "3", "expectedVariableNames": [], "unitTests": [], "checkVariableNames": false, "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "showCorrectAnswer": true, "showFeedbackIcon": true, "showPreview": true, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "vsetRange": [0, 1], "failureRate": 1, "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true, "answer": "({f}s^2+({f}*{a}+{g}+{d}*{f})s+(({g}+{a}*{f})*{d}+{c}))/((s+{d})(s^2+{a}s}+{b}))"}], "sortAnswers": false, "variableReplacements": [], "scripts": {}, "extendBaseMarkingAlgorithm": true}], "extensions": [], "functions": {}, "type": "question", "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"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": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"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/"}]}