// Numbas version: exam_results_page_options
{"name": "Shaheen's copy of 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": [{"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}*lambda=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>", "functions": {}, "parts": [{"displayColumns": 1, "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>", "type": "1_n_2", "showCorrectAnswer": true, "distractors": ["", "", ""], "scripts": {}, "shuffleChoices": true, "minMarks": 0, "choices": ["<p>{correctform}</p>", "<p>{incorrectform[0]}</p>", "<p>{incorrectform[1]}</p>"], "maxMarks": 0, "marks": 0, "matrix": [1, 0, 0], "displayType": "radiogroup"}, {"gaps": [{"vsetrange": [0, 1], "vsetrangepoints": 5, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "expectedvariablenames": [], "answer": "{lambda1}", "scripts": {}, "checkvariablenames": false, "marks": 1, "checkingtype": "absdiff", "showpreview": true}, {"vsetrange": [0, 1], "vsetrangepoints": 5, "checkingaccuracy": 0.001, "type": "jme", "showCorrectAnswer": true, "expectedvariablenames": [], "answer": "{lambda2}", "scripts": {}, "checkvariablenames": false, "marks": 1, "checkingtype": "absdiff", "showpreview": true}], "scripts": {}, "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>", "marks": 0, "type": "gapfill", "showCorrectAnswer": true}], "variables": {"disc": {"definition": "b1^2-4*a1*c1", "name": "disc", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "correctform": {"definition": "switch (\n      disc>0, forms[0],\n      disc=0, forms[1],\n      disc<0, forms[2]\n    )", "name": "correctform", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "b1": {"definition": "random(-9..9 except 0)", "name": "b1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "forms": {"definition": "[\"$A\\\\mathrm{e}^{\\\\lambda_1 x}+B\\\\mathrm{e}^{\\\\lambda_2 x}$\",\"$(A+Bx)\\\\mathrm{e}^{\\\\lambda x}$\",\"$\\\\mathrm{e}^{\\\\alpha x}\\\\biggl(A\\\\cos(\\\\beta x)+B\\\\sin(\\\\beta x)\\\\biggr)$\"]", "name": "forms", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a1": {"definition": "random(1..9)", "name": "a1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "incorrectform": {"definition": "switch (\n      disc>0, [forms[1],forms[2]],\n      disc=0, [forms[0],forms[2]],\n      disc<0, [forms[0],forms[1]]\n    )", "name": "incorrectform", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "lambda1": {"definition": "precround((-b1+sqrt(disc))/(2*a1),3)", "name": "lambda1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "lambda2": {"definition": "precround((-b1-sqrt(disc))/(2*a1),3)", "name": "lambda2", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "c1": {"definition": "random(-9..9 except 0)", "name": "c1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "ltgteq": {"definition": "switch (\n      disc>0, \"greater than\",\n      disc=0, \"equal to\",\n      disc<0, \"less than\"\n    )", "name": "ltgteq", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "correctform", "incorrectform", "forms", "disc", "b1", "c1", "lambda1", "lambda2", "ltgteq"], "statement": "<p>You are given the differential equation</p>\n  <p>\\[\\simplify{{a1}*y''+{b1}*y'+{c1}*y=0}.\\]</p>", "rulesets": {}, "name": "Shaheen's copy of Solve a constant coefficient second order ODE, ", "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "type": "question", "tags": ["checked2015", "MAS1603", "MAS2105"], "metadata": {"description": "<p>Find the solution of a constant coefficient second order ordinary differential equation of the form $ay''+by'+cy=0$.</p>", "notes": "", "licence": "Creative Commons Attribution 4.0 International"}, "preamble": {"css": "", "js": ""}, "variable_groups": [], "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}]}]}], "contributors": [{"name": "Shaheen Charlwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1819/"}]}