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