// Numbas version: finer_feedback_settings
{"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": [{"variable_groups": [], "variables": {"tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.001", "name": "tol", "description": ""}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "c1", "description": ""}, "a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(1/2*(c1+d1*(sqrt(a1/b1))),3)", "name": "a", "description": ""}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "b1", "description": ""}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(1/2*(c1-d1*(sqrt(a1/b1))),3)", "name": "b", "description": ""}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "name": "a1", "description": ""}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "name": "d1", "description": ""}}, "ungrouped_variables": ["a", "b", "a1", "b1", "tol", "c1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Solve a constant coefficient second order ODE, ", "showQuestionGroupNames": false, "functions": {}, "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"answer": "sqrt({b1})/sqrt({a1})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "all", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "-sqrt({b1})/sqrt({a1})", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "answersimplification": "all", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Solve the differential equation, and enter the values of $\\lambda_1$ and $\\lambda_2$ 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>$\\lambda_1=$ [[0]]</p>\n<p>$\\lambda_2=$ [[1]]</p>", "variableReplacements": [], "marks": 0}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "A-tol", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "maxValue": "A+tol"}, {"showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "scripts": {}, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "B-tol", "showCorrectAnswer": true, "variableReplacements": [], "marks": 1, "maxValue": "B+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Now find the 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>Enter your answers to 3d.p.</p>\n<p>$A=$ [[0]]</p>\n<p>$B=$ [[1]]</p>", "variableReplacements": [], "marks": 0}], "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\\mathrm{e}^{\\lambda_1 x}+B\\mathrm{e}^{\\lambda_2 x},\\]</p>\n    <p>where $A,B,\\lambda_1,\\lambda_2\\in\\mathbb{R}$ and $\\lambda_1&gt;\\lambda_2$.</p>", "tags": ["checked2015", "MAS1603", "MAS2105"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''-by=0$. Distinct roots.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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})}.\\]</p>\n<p>The general solution to the differential equation, therefore, is</p>\n<p>\\[y(x)=A\\exp\\left(\\simplify{sqrt({b1})/sqrt({a1})*x}\\right)+B\\exp\\left(\\simplify{-sqrt({b1})/sqrt({a1})*x}\\right).\\]</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+B\\]</p>\n<p>and</p>\n<p>\\[\\var{d1}=y'(0)=\\simplify{A*(sqrt({b1})/sqrt({a1}))-B*(sqrt({b1})/sqrt({a1}))}.\\]</p>\n<p>Some rearrangement then gives</p>\n<p>\\[A=\\simplify{1/2*({c1}+{d1}*(sqrt({a1})/sqrt({b1})))}=\\var{A}\\;\\text{to 3 d.p.,}\\]</p>\n<p>and</p>\n<p>\\[B=\\simplify{1/2*({c1}-{d1}*(sqrt({a1})/sqrt({b1})))}=\\var{B}\\;\\text{to 3 d.p.}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}