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{"name": "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": [{"tags": [], "parts": [{"prompt": "<p>\\(Q(s)=\\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "marks": 0, "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": [], "gaps": [{"expectedvariablenames": [], "showFeedbackIcon": true, "checkvariablenames": false, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "answer": "({f}s^2+({f}*{a}+{g}+{d}*{f})s+(({g}+{a}*{f})*{d}+{c}))/((s+{d})(s^2+{a}s}+{b}))", "scripts": {}, "variableReplacements": [], "marks": "3", "vsetrangepoints": 5, "type": "jme", "checkingtype": "absdiff", "showCorrectAnswer": true, "showpreview": true}]}], "name": "Year 3 Find the Laplace transfrom of ode", "functions": {}, "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "variable_groups": [], "rulesets": {}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g"], "preamble": {"js": "", "css": ""}, "advice": "<p>\\(\\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>\\(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>\\(s^2Q(s)-\\var{f}s-\\var{g}+\\var{a}sQ(s)-\\var{a}*\\var{f}+\\var{b}Q(s)=\\frac{\\var{c}}{s+\\var{d}}\\)</p>\n<p></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>\\((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>\\(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>", "variables": {"d": {"definition": "random(3..8#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "d"}, "f": {"definition": "random(1..6#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "f"}, "b": {"definition": "random(10..25#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "b"}, "a": {"definition": "random(2..10#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "a"}, "c": {"definition": "random(3..12#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "c"}, "g": {"definition": "random(3..9#1)", "description": "", "templateType": "randrange", "group": "Ungrouped variables", "name": "g"}}, "statement": "<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>", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}