// Numbas version: exam_results_page_options
{"name": "Differential Equations:  Second Order Repeated Roots", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differential Equations:  Second Order Repeated Roots", "tags": [], "metadata": {"description": "<p>Solve:&nbsp;$\\displaystyle \\frac{d^2y}{dx^2}+2a\\frac{dy}{dx}+a^2y=0,\\;y(0)=c$ and $y(1)=d$. &nbsp;(Equal roots example).</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Find the solution of the differential equation</strong>:</p>\n<p><br/>\\[\\dfrac{d^2y}{dx^2}+\\var{2*a}\\dfrac{dy}{dx}+\\var{a^2}y=0\\]</p>\n<p><br/>which satisfies $y(0)=\\var{c}$ and $y(1)=\\var{d}$.</p>", "advice": "<p>The auxillary equation is $\\simplify[std]{lambda^2+{2*a}lambda+{a^2}}=0$.</p>\n  <p>On solving this equation we get $\\lambda=\\var{-a}$ twice.</p>\n  <p>Hence the general solution is:<br/>\\[y = \\simplify[std]{A*e^({-a}x)+B*x*e^({-a}x)}\\]<br/>The boundary conditions give:</p>\n  <p>$y(0)=\\var{c} \\Rightarrow A=\\var{c}$</p>\n  <p>$y(1)=\\var{d} \\Rightarrow \\simplify{Ae^{-a}+Be^{-a}={d}}\\Rightarrow A+B = \\simplify{{d}e^{a}}$</p>\n  <p>So $B=\\simplify{{d}e^{a}-{c}}=\\var{f1}$ to 3 decimal places.</p>\n  <p>Hence the solution is:<br/>\\[y=\\simplify{(({c} * Exp(({( - a)} * x))) + ({f1} * x * Exp(({( - a)} * x))))}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "s*random(3..7)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything"}, "f": {"name": "f", "group": "Ungrouped variables", "definition": "d*exp(a)-c", "description": "", "templateType": "anything"}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "templateType": "anything"}, "f1": {"name": "f1", "group": "Ungrouped variables", "definition": "precround(f,3)", "description": "", "templateType": "anything"}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "f1", "c", "b", "d", "f", "s"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Solution is:</p>\n      <p>$y=\\;\\;$[[0]]</p>\n      <p>Input all numbers correct to 3 decimal places.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 3, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{c} * Exp({ - a} * x) + {f1} * x * Exp({- a} * x)", "answerSimplification": "std,!fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 1e-05, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}