// Numbas version: exam_results_page_options
{"name": "Differential equation with a repeated linear factor: Algebraic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": [], "name": "Differential equation with a repeated linear factor: Algebraic", "tags": [], "statement": "<p>Solve the differential equation:</p>\n<p>&nbsp; &nbsp; &nbsp;\\(\\frac{d^2i}{dt^2}+\\simplify{{a1}+{a1}}\\frac{di}{dt}+\\simplify{{a1}*{a1}}i(t)=\\var{c1}e^{-\\var{d1}t}\\) &nbsp;where \\(i(0)=\\var{i0} \\,\\, and \\,\\, &nbsp;i'(0)=\\var{i1}\\)</p>\n<p></p>\n<p></p>", "variables": {"f4": {"definition": "((-{d1}+{a1})(-{d1}+{a1}))", "description": "", "name": "f4", "group": "Ungrouped variables", "templateType": "anything"}, "f1": {"definition": "{c1}", "description": "", "name": "f1", "group": "Ungrouped variables", "templateType": "anything"}, "i1": {"definition": "random(1..10) ", "description": "", "name": "i1", "group": "Ungrouped variables", "templateType": "anything"}, "c1": {"definition": "random(1..10)", "description": "", "name": "c1", "group": "Ungrouped variables", "templateType": "anything"}, "f3": {"definition": "{i0}*({a1}-{d1})*({a1}-{d1})-{c1}", "description": "", "name": "f3", "group": "Ungrouped variables", "templateType": "anything"}, "d1": {"definition": "random(12..16)", "description": "", "name": "d1", "group": "Ungrouped variables", "templateType": "anything"}, "f5": {"definition": "({c1}+(-{i0}*{a1}+{i1}+2*{a1}*{i0})*(-{a1}+{d1}))", "description": "", "name": "f5", "group": "Ungrouped variables", "templateType": "anything"}, "f6": {"definition": "(-{a1}+{d1})", "description": "", "name": "f6", "group": "Ungrouped variables", "templateType": "anything"}, "i0": {"definition": "random(1..10)", "description": "", "name": "i0", "group": "Ungrouped variables", "templateType": "anything"}, "f2": {"definition": "((-{d1}+{a1})(-{d1}+{a1}))", "description": "", "name": "f2", "group": "Ungrouped variables", "templateType": "anything"}, "a1": {"definition": "random(1..5)", "description": "", "name": "a1", "group": "Ungrouped variables", "templateType": "anything"}}, "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "gaps": [{"checkingtype": "absdiff", "checkvariablenames": false, "scripts": {}, "expectedvariablenames": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "showpreview": true, "checkingaccuracy": 0.001, "type": "jme", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "marks": "3", "answer": "{f1}/{f2}e^(-{d1}t)+{f3}/{f4}e^(-{a1}t)+{f5}/{f6}te^(-{a1}t)", "vsetrange": [0, 1], "showCorrectAnswer": true}], "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "prompt": "<p>\\(i(t)=\\)&nbsp;[[0]]</p>", "showCorrectAnswer": true}], "rulesets": {}, "functions": {}, "ungrouped_variables": ["a1", "c1", "d1", "i0", "i1", "f1", "f2", "f3", "f4", "f5", "f6"], "preamble": {"js": "", "css": ""}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solve a&nbsp;Differential equation&nbsp;with a repeated linear factor</p>"}, "advice": "<p>\\(\\frac{d^2i}{dt^2}+\\simplify{{a1}+{a1}}\\frac{di}{dt}+\\simplify{{a1}*{a1}}i(t)=\\var{c1}e^{-\\var{d1}t}\\) &nbsp;where \\(i(0)=\\var{i0} \\,\\, and \\,\\, &nbsp;i'(0)=\\var{i1}\\)</p>\n<p>The Laplace transform of this is given by:</p>\n<p>\\(s^2I(s)-\\var{i0}s-\\var{i1}+\\simplify{{a1}+{a1}}(sI(s)-\\var{i0})+\\simplify{{a1}*{a1}}I(s)=\\frac{\\var{c1}}{s+\\var{d1}}\\)</p>\n<p>Gathering only \\(I(s)\\) terms on the left hand side and factoring gives:</p>\n<p>\\((s^2+\\simplify{{a1}+{a1}}s+\\simplify{{a1}*{a1}})I(s)=\\frac{\\var{c1}}{s+\\var{d1}}+\\var{i0}s+\\simplify{{i1}+({a1}+{a1})*{i0}}\\)</p>\n<p>\\((s^2+\\simplify{{a1}+{a1}}s+\\simplify{{a1}*{a1}})I(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{a1})*{i0})*(s+{d1})}}{s+\\var{d1}}\\)</p>\n<p>\\(I(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{a1})*{i0})*(s+{d1})}}{(s+\\var{d1})(s+\\var{a1})^2}\\)</p>\n<p>\\(I(s)=\\frac{A}{s+\\var{d1}}+\\frac{B}{s+\\var{a1}}+\\frac{C}{(s+\\var{a1})^2}\\)</p>\n<p>\\(\\simplify{{c1}+({i0}s+{i1}+({a1}+{a1})*{i0})*(s+{d1})}=A(s+\\var{a1})(s+\\var{a1})+B(s+\\var{d1})(s+\\var{a1})+C(s+\\var{d1})\\)</p>\n<p>Let \\(s=-\\var{d1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+({i0}*-{d1}+{i1}+({a1}+{a1})*{i0})*(-{d1}+{d1})}=\\simplify{(-{d1}+{a1})(-{d1}+{a1})}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{({c1})/((-{d1}+{a1})(-{d1}+{a1}))}\\)</p>\n<p>Let \\(s=-\\var{a1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1})}=\\simplify{(-{a1}+{d1})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1}))}\\)</p>\n<p>Compare the coefficients of \\(s^2\\) &nbsp;&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(\\var{i0}=A+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{{i0}-(({c1})/((-{d1}+{a1})(-{d1}+{a1})))}\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{({i0}*(-{d1}+{a1})(-{d1}+{a1})-{c1})/((-{d1}+{a1})(-{d1}+{a1}))}\\)</p>\n<p>\\(i(t)=\\simplify{({c1})/((-{d1}+{a1})(-{d1}+{a1}))}e^{-\\var{d1}t}+\\simplify{({i0}*(-{d1}+{a1})(-{d1}+{a1})-{c1})/((-{d1}+{a1})(-{d1}+{a1}))}e^{-\\var{a1}t}+\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{a1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1}))}te^{-\\var{a1}t}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "variablesTest": {"maxRuns": "222", "condition": ""}, "variable_groups": [], "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/"}]}