// 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": {"correctform": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n      disc>0, forms[0],\n      disc=0, forms[1],\n      disc<0, forms[2]\n    )", "description": "", "name": "correctform"}, "ltgteq": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n      disc>0, \"greater than\",\n      disc=0, \"equal to\",\n      disc<0, \"less than\"\n    )", "description": "", "name": "ltgteq"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "c1"}, "disc": {"group": "Ungrouped variables", "templateType": "anything", "definition": "b1^2-4*a1*c1", "description": "", "name": "disc"}, "incorrectform": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch (\n      disc>0, [forms[1],forms[2]],\n      disc=0, [forms[0],forms[2]],\n      disc<0, [forms[0],forms[1]]\n    )", "description": "", "name": "incorrectform"}, "b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(-9..9 except 0)", "description": "", "name": "b1"}, "lambda2": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround((-b1-sqrt(disc))/(2*a1),3)", "description": "", "name": "lambda2"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)", "description": "", "name": "a1"}, "lambda1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround((-b1+sqrt(disc))/(2*a1),3)", "description": "", "name": "lambda1"}, "forms": {"group": "Ungrouped variables", "templateType": "anything", "definition": "[\"$A\\\\mathrm{e}^{\\\\lambda_1 x}+B\\\\mathrm{e}^{\\\\lambda_2 x}$\",\"$(A+Bx)\\\\mathrm{e}^{\\\\lambda x}$\",\"$\\\\mathrm{e}^{\\\\alpha x}\\\\biggl(A\\\\cos(\\\\beta x)+B\\\\sin(\\\\beta x)\\\\biggr)$\"]", "description": "", "name": "forms"}}, "ungrouped_variables": ["a1", "correctform", "incorrectform", "forms", "disc", "b1", "c1", "lambda1", "lambda2", "ltgteq"], "name": "Solve a constant coefficient second order ODE, ", "functions": {}, "preamble": {"css": "", "js": ""}, "parts": [{"displayType": "radiogroup", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showCellAnswerState": true, "choices": ["<p>{correctform}</p>", "<p>{incorrectform[0]}</p>", "<p>{incorrectform[1]}</p>"], "prompt": "<p>Which of the following choices defines the form of the general solution of the differential equation?</p>\n<p>In each case $A$ and $B$ are arbitrary constants, and $\\lambda_1$, $\\lambda_2$, $\\lambda$, $\\alpha$, and $\\beta$ are other constants arising from the solution of the auxiliary equation (their actual values are not important for this part of the question).</p>", "distractors": ["", "", ""], "matrix": [1, 0, 0], "unitTests": [], "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "displayColumns": 1, "showCorrectAnswer": true, "variableReplacements": [], "marks": 0}, {"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Find the general solution of the differential equation, by setting up an appropriate auxiliary equation, solving it, and entering the solutions &nbsp;$\\lambda_1$ and $\\lambda_2$ &nbsp;of the auxiliary equation in the boxes. &nbsp;If the solutions are real and distinct, enter the greatest solution as $\\lambda_1$; if the solutions are repeated, enter the same values for $\\lambda_1$ and $\\lambda_2$; if the solutions are complex, enter the solution with the greatest imaginary part as $\\lambda_1$.</p>\n<p>Enter your answers to 3 d.p.</p>\n<p>$\\lambda_1=$ [[0]]</p>\n<p>$\\lambda_2=$ [[1]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{lambda1}", "showCorrectAnswer": true, "failureRate": 1, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}, {"answer": "{lambda2}", "showCorrectAnswer": true, "failureRate": 1, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "showPreview": true, "checkVariableNames": false, "checkingType": "absdiff", "vsetRange": [0, 1], "type": "jme", "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "expectedVariableNames": [], "unitTests": [], "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "statement": "<p>You are given the differential equation</p>\n  <p>\\[\\simplify{{a1}*y''+{b1}*y'+{c1}*y=0}.\\]</p>", "tags": ["checked2015"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by'+cy=0$.</p>"}, "type": "question", "extensions": [], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>To determine the form of the general solution of the equation</p>\n<p>\\[ay''+by'+c=0,\\]</p>\n<p>first set $y=\\mathrm{e}^{\\lambda x}$, and substitute to obtain</p>\n<p>\\[a\\lambda^2+b\\lambda+c=0,\\]</p>\n<p>for which the solutions are</p>\n<p>\\[\\lambda_{1,2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\\]</p>\n<p>If $b^2-4ac&gt;0$, then the roots are real and distinct, and the solution takes the form</p>\n<p>\\[y(x)=\\var{forms[0]}.\\]</p>\n<p>If $b^2-4ac=0$, then the roots are real and repeated, and the solution takes the form</p>\n<p>\\[y(x)=\\var{forms[1]}.\\]</p>\n<p>If $b^2-4ac&lt;0$, then the roots are complex, and the solution takes the form</p>\n<p>\\[y(x)=\\var{forms[2]},\\]</p>\n<p>where $\\lambda_1=\\alpha+i\\beta$ and $\\lambda_2=\\alpha-i\\beta$.</p>\n<h4>a)</h4>\n<p>In this question we have $\\simplify{{a1}*y''+{b1}*y'+{c1}*y=0}$, and then</p>\n<p>\\[b^2-4ac=\\var{b1^2}-4\\times(\\var{a1*c1})=\\var{disc},\\]</p>\n<p>which is {ltgteq} zero, so the general solution takes the form</p>\n<p>\\[y(x)=\\var{correctform}.\\]</p>\n<h4>b)</h4>\n<p>Making the substitution $y=\\mathrm{e}^{\\lambda x}$, then gives</p>\n<p>\\[\\simplify{{a1}*lambda^2+{b1}*lambda+{c1}=0},\\]</p>\n<p>which has solutions</p>\n<p>\\[\\lambda_1=\\frac{\\var{-b1}+\\sqrt{\\var{disc}}}{\\var{2*a1}}=\\var{lambda1} \\text{ to 3 d.p.,}\\]</p>\n<p>and</p>\n<p>\\[\\lambda_2=\\frac{\\var{-b1}-\\sqrt{\\var{disc}}}{\\var{2*a1}}=\\var{lambda2} \\text{ to 3 d.p.}\\]</p>", "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/"}]}