// Numbas version: exam_results_page_options
{"name": "Differential equation with a simple quadratic factor: Algebraic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "<p>Solve the differential equation:</p>\n<p>&nbsp; &nbsp; &nbsp;\\(\\frac{d^2i}{dt^2}+\\simplify{{a1}^2}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>", "variablesTest": {"maxRuns": "222", "condition": ""}, "variable_groups": [], "metadata": {"description": "<p>Solve a&nbsp;Differential equation&nbsp;with a repeated linear factor</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a1", "c1", "d1", "i0", "i1", "f1", "f2", "f3", "f4", "f5", "f6"], "tags": [], "name": "Differential equation with a simple quadratic factor: Algebraic", "functions": {}, "advice": "<p></p>\n<p>&nbsp;\\(\\frac{d^2i}{dt^2}+\\simplify{{a1}^2}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}^2}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}^2})I(s)=\\frac{\\var{c1}}{s+\\var{d1}}+\\var{i0}s+\\var{i1}\\)</p>\n<p>\\((s^2+\\simplify{{a1}*{a1}})I(s)=\\frac{\\simplify{{c1}+({i0}s+{i1})*(s+{d1})}}{s+\\var{d1}}\\)</p>\n<p>\\(I(s)=\\frac{\\simplify{{c1}+({i0}s+{i1})*(s+{d1})}}{(s+\\var{d1})(s^2+\\simplify{{a1}^2})}\\)</p>\n<p>\\(I(s)=\\frac{A}{s+\\var{d1}}+\\frac{Bs+c}{s^2+\\simplify{{a1}^2}}\\)</p>\n<p>\\(\\simplify{{c1}+({i0}s+{i1})*(s+{d1})}=A(s^2+\\simplify{{a1}^2})+Bs(s+\\var{d1})+c(s+\\var{d1})\\)</p>\n<p>Let \\(s=-\\var{d1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{c1}=\\simplify{{d1}^2+{a1}^2}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{({c1})/(({d1}^2+{a1}^2))}\\)</p>\n<p>Let \\(s=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+{i1}*{d1}}=\\simplify{{a1}^2}A+{\\var{d1}}c\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{c1}+{i1}*{d1}}=\\simplify{{a1}^2*{c1}/({d1}^2+{a1}^2)}+{\\var{d1}}c\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{d1}c=\\frac{\\simplify{{i1}*{d1}*({d1}^2+{a1}^2)+{c1}*{d1}^2-{a1}}}{\\simplify{({d1}^2+{a1}^2)}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(c=\\frac{\\simplify{{i1}*({d1}^2+{a1}^2)+{c1}*{d1}}}{\\simplify{({d1}^2+{a1}^2)}}\\)</p>\n<p style=\"padding-left: 30px;\"></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=\\var{i0}-\\simplify{({c1})/(({d1}^2+{a1}^2))}=\\simplify{(({d1}^2+{a1}^2)*{i0}-{c1})/({d1}^2+{a1}^2)}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p>\\(I(s)=\\frac{\\simplify{{f1}/{f2}}}{s+\\var{d1}}+\\frac{\\simplify{{f3}/{f4}}s}{s^2++\\simplify{{a1}^2})}+\\frac{\\simplify{{a1}*{f5}/{f6}}}{s^2+\\simplify{{a1}^2})}\\)</p>\n<p>\\(I(s)=\\frac{\\var{f1}}{\\var{f2}}e^{-\\var{d1}t}+\\frac{\\var{f3}}{\\var{f4}}cos(\\var{a1}t)+\\frac{\\var{f5}}{\\var{f6}}sin(\\var{a1}t)\\)</p>", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "variableReplacements": [], "vsetrangepoints": 5, "showCorrectAnswer": true, "vsetrange": [0, 1], "showFeedbackIcon": true, "marks": "3", "checkingaccuracy": 0.001, "checkingtype": "absdiff", "expectedvariablenames": [], "answer": "{f1}/{f2}e^(-{d1}t)+{f3}/{f4}cos({a1}t)+{f5}/{f6}sin({a1}t)", "scripts": {}, "showpreview": true, "checkvariablenames": false, "type": "jme"}], "showCorrectAnswer": true, "prompt": "<p>\\(i(t)=\\)&nbsp;[[0]]</p>", "showFeedbackIcon": true, "marks": 0, "type": "gapfill"}], "variables": {"c1": {"name": "c1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1..10)"}, "i0": {"name": "i0", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1..10)"}, "a1": {"name": "a1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..7)"}, "d1": {"name": "d1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(2..11)"}, "f5": {"name": "f5", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{i1}*({a1}^2+{d1}^2)+{c1}*{d1}"}, "i1": {"name": "i1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "random(1..10) "}, "f4": {"name": "f4", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(({d1}*{d1})+({a1}*{a1}))"}, "f2": {"name": "f2", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{d1}^2+{a1}^2"}, "f6": {"name": "f6", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "(({d1}*{d1})+({a1}*{a1}))*{a1}"}, "f1": {"name": "f1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{c1}"}, "f3": {"name": "f3", "templateType": "anything", "description": "", "group": "Ungrouped variables", "definition": "{i0}*({a1}^2+{d1}^2)-{c1}"}}, "extensions": [], "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/"}]}