// Numbas version: exam_results_page_options
{"name": "Solve a second order ODE with repeated roots, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..7)", "description": "", "name": "b"}, "f1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(f,3)", "description": "", "name": "f1"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s"}, "d": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "description": "", "name": "d"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..6)", "description": "", "name": "c"}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..7)", "description": "", "name": "a"}, "f": {"group": "Ungrouped variables", "templateType": "anything", "definition": "d*exp(a)-c", "description": "", "name": "f"}}, "ungrouped_variables": ["a", "f1", "c", "b", "d", "f", "s"], "name": "Solve a second order ODE with repeated roots, ", "functions": {}, "parts": [{"customMarkingAlgorithm": "", "showCorrectAnswer": true, "prompt": "<p>Solution is:</p>\n      <p>$y=\\;\\;$[[0]]</p>\n      <p>Input all numbers correct to 3 decimal places.</p>", "unitTests": [], "showFeedbackIcon": true, "scripts": {}, "gaps": [{"answer": "{c} * Exp({ - a} * x) + {f1} * x * Exp({- a} * x)", "showCorrectAnswer": true, "checkingAccuracy": 1e-05, "customMarkingAlgorithm": "", "answerSimplification": "std,!fractionNumbers", "expectedVariableNames": [], "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "vsetRangePoints": 5, "showFeedbackIcon": true, "scripts": {}, "marks": 3, "type": "jme", "unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "failureRate": 1}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>Find the solution of:<br/>\\[\\simplify[std]{(d^2y/(d*x^2))+{2*a}*((d*y/(d*x)))+{a^2}y}=0\\]<br/>which satisfies $y(0)=\\var{c}$ and $y(1)=\\var{d}$.</p>", "tags": ["2nd order differential equation", "auxillary equation", "auxillary equation with equal roots", "boundary conditions", "calculus", "Calculus", "checked2015", "constant coefficients", "differential equations", "Differential equations", "equal roots", "exponential function", "general solution", "linear differential equations", "linear differential equations with constant coefficients", "ODE", "ode", "quadratic function", "repeated roots for auxillary equation", "second order differential equations", "solving differential equations", "solving quadratic function", "trigonometric functions"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "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>", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "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>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}