// Numbas version: exam_results_page_options
{"name": "Harry's copy of 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": [{"type": "question", "variable_groups": [], "variables": {"d": {"definition": "random(1..6)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "d"}, "f": {"definition": "d*exp(a)-c", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "f"}, "b": {"definition": "random(1..7)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "b"}, "f1": {"definition": "precround(f,3)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "f1"}, "a": {"definition": "s*random(1..7)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "a"}, "s": {"definition": "random(1,-1)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "s"}, "c": {"definition": "random(1..6)", "description": "", "group": "Ungrouped variables", "templateType": "anything", "name": "c"}}, "statement": "<p>Find the solution of:<br/>\\[\\simplify[std]{(d^2y/dx^2)+{2*a}*(dy/dx)+{a^2}y}=0\\]<br/>which satisfies $y(0)=\\var{c}$ and $y(1)=\\var{d}$.</p>", "showQuestionGroupNames": false, "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>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"type": "gapfill", "scripts": {}, "gaps": [{"type": "jme", "answer": "{c} * Exp({ - a} * x) + {f1} * x * Exp({- a} * x)", "marks": 3, "showpreview": true, "vsetrangepoints": 5, "checkingtype": "absdiff", "checkvariablenames": false, "expectedvariablenames": [], "scripts": {}, "vsetrange": [0, 1], "showCorrectAnswer": true, "answersimplification": "std,!fractionNumbers", "checkingaccuracy": 1e-05}], "showCorrectAnswer": true, "prompt": "<p>Solution is:</p>\n      <p>$y=\\;\\;$[[0]]</p>\n      <p>Input all numbers correct to 3 decimal places.</p>", "marks": 0}], "preamble": {"js": "", "css": ""}, "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", "MAS1603", "MAS2106", "ODE", "ode", "quadratic function", "repeated roots for auxillary equation", "second order differential equations", "solving differential equations", "solving quadratic function", "trigonometric functions"], "functions": {}, "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n            <p>Added and edited tags.</p>\n            <p>Improved display in prompt.</p>\n            <p>Checked calculation.</p>\n            <p><strong>18/07/2012:</strong></p>\n            <p>Added description.</p>\n            <p><strong>23/07/2012:</strong></p>\n            <p>Added tags.</p>\n            <p></p>\n            <p>Question appears to be working correctly.</p>\n            <p>&nbsp;</p>", "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>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "f1", "c", "b", "d", "f", "s"], "variablesTest": {"maxRuns": 100, "condition": ""}, "question_groups": [{"pickingStrategy": "all-ordered", "pickQuestions": 0, "name": "", "questions": []}], "name": "Harry's copy of Solve a second order ODE with repeated roots, ", "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}