// Numbas version: exam_results_page_options
{"name": "Solve a constant coefficient second order ODE ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solve a constant coefficient second order ODE ", "tags": ["checked2015"], "metadata": {"description": "<p>Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by=0$. Complex roots.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The general solution of the differential equation</p>\n    <p>\\[\\simplify{{a1}*y''+{b1}*y}=0\\]</p>\n    <p>can be written in the form</p>\n    <p>\\[y(x)=A\\cos(\\lvert\\lambda_1\\rvert x)+B\\sin(\\lvert\\lambda_2\\rvert x),\\]</p>\n    <p>where $A,B\\in\\mathbb{R}$, and $\\lambda_1,\\lambda_2\\in\\mathbb{C}$.</p>", "advice": "<h4>a)</h4>\n<p>We can solve the differential equation by making the assumption that $y=\\mathrm{e}^{\\lambda x}$.</p>\n<p>By substituting this expression for $y$ into the equation, and cancelling terms in $\\mathrm{e}^{\\lambda x}$ (which we can do, because $\\mathrm{e}^{\\lambda x}\\ne 0$), we obtain</p>\n<p>\\[\\simplify{{a1}*lambda^2+{b1}}=0,\\]</p>\n<p>which has solutions</p>\n<p>\\[\\lambda_{1,2}=\\pm\\simplify{sqrt({b1}/{a1})}i.\\]</p>\n<p>The general solution to the differential equation, therefore, is</p>\n<p>\\[y(x)=A\\cos\\left(\\simplify{sqrt({b1})/sqrt({a1})*x}\\right)+B\\sin\\left(\\simplify{sqrt({b1})/sqrt({a1})*x}\\right).\\]</p>\n<p>This is the only form of the solution accepted in this question, but note that the general solution could also be written as</p>\n<p>\\[y(x)=A\\cos\\left(\\simplify{sqrt({b1})/sqrt({a1})*x}\\right)+B\\sin\\left(\\simplify{-sqrt({b1})/sqrt({a1})*x}\\right),\\]</p>\n<p>or</p>\n<p>\\[y(x)=A\\cos\\left(\\simplify{-sqrt({b1})/sqrt({a1})*x}\\right)+B\\sin\\left(\\simplify{sqrt({b1})/sqrt({a1})*x}\\right).\\]</p>\n<p>These forms just redefine what is meant by the constants $A$ and $B$, however, since</p>\n<p>\\[\\cos(-x)=\\cos(x)\\quad\\text{and}\\quad\\sin(-x)=-\\sin(x).\\]</p>\n<h4>b)</h4>\n<p>Applying the conditions $y(0)=\\var{c1}$ and $y'(0)=\\var{d1}$ gives</p>\n<p>\\[\\var{c1}=y(0)=A\\]</p>\n<p>and</p>\n<p>\\[\\var{d1}=y'(0)=\\simplify{B*(sqrt({b1})/sqrt({a1}))}.\\]</p>\n<p>So $A=\\var{c1}$, and $B=\\simplify{{d1}*sqrt({a1}/{b1})}$.</p>", "rulesets": {}, "extensions": [], "variables": {"b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything"}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "c1", "b1", "d1"], "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>Solve the differential equation, and enter the values of $\\lvert\\lambda_1\\rvert$ and $\\lvert\\lambda_2\\rvert$ in the boxes.</p>\n<p>Do not enter decimals in your answers. If you need to enter a square root, e.g. $\\sqrt{x}$, enter this as <code>sqrt(x)</code>.</p>\n<p>$\\lvert\\lambda_1\\rvert=$ [[0]]</p>\n<p>$\\lvert\\lambda_2\\rvert=$ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "sqrt({b1})/sqrt({a1})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not enter decimals in your answer.</p>"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "sqrt({b1})/sqrt({a1})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not enter decimals in your answer.</p>"}, "valuegenerators": []}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Now, using the general form of solution as shown above, with $\\lvert\\lambda_1\\rvert$ and $\\lvert\\lambda_2\\rvert$, find the exact solution that satisfies the conditions $y(0)=\\var{c1}$ and $y'(0)=\\var{d1}$, by calculating the values of the constants $A$ and $B$, and entering them in the boxes.</p>\n<p>Do not enter decimals in your answers. If you need to enter a square root, e.g. $\\sqrt{x}$, enter this as&nbsp;<code>sqrt(x)</code>.</p>\n<p>$A=$ [[0]]</p>\n<p>$B=$ [[1]]</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{c1}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not enter decimals in your answer.</p>"}, "valuegenerators": []}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{d1}*sqrt({a1})/sqrt({b1})", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Do not enter decimals in your answer.</p>"}, "valuegenerators": []}], "sortAnswers": false}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}