// Numbas version: exam_results_page_options
{"name": "Second order ODE with constant coefficients and boundary conditions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"a": {"group": "Ungrouped variables", "templateType": "anything", "definition": "s*random(1..5)", "description": "", "name": "a"}, "b": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..7)", "description": "", "name": "b"}, "c": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..5)", "description": "", "name": "c"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1,-1)", "description": "", "name": "s"}}, "ungrouped_variables": ["a", "s", "b", "c"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Second order ODE with constant coefficients and boundary conditions", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"answer": "exp({- a} * x) * (cos({b} * x) + ({c+a} / {b}) * sin({b} * x))", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 1e-05, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "answersimplification": "std", "marks": 3, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Solution is:</p>\n      <p>$y=\\;\\;$[[0]]</p>\n      <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Solve:<br/>\\[\\simplify[std]{(d^2y/dx^2)+{2*a}*(dy/dx)+{a^2+b^2}y}=0\\]<br/>which satisfies $y(0)=1$ and $y'(0)=\\var{c}$ (where prime denotes the derivative).</p>", "tags": ["2nd order differential equation", "auxiliary equation", "boundary conditions on differential equation", "Calculus", "checked2015", "complex roots of auxillary equation", "constant coefficients", "differential equation", "differential equation ", "exponential function", "finding the auxillary equation", "linear differential equation", "MAS1603", "ode", "quadratic equation", "second order differential equation", "solving differential equations", "solving quadratic equation", "trigonometric functions"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>29/06/2012:</strong></p>\n    <p><strong><br /></strong>Added tags. Edited tags.</p>\n    <p>Improved display.</p>\n    <p>Checked answer.</p>\n    <p><strong>23/07/2012:</strong></p>\n    <p>Added tags.</p>\n    <p>Question appears to be working correctly.</p>\n    <p><strong>04/11/2012:</strong></p>\n    <p><strong><br /></strong>Corrected mistake in solution.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Solve:&nbsp;$\\displaystyle \\frac{d^2y}{dx^2}+2a\\frac{dy}{dx}+(a^2+b^2)y=0,\\;y(0)=1$ and $y'(0)=c$.&nbsp;</p>"}, "advice": "<p>The auxillary equation is $\\simplify[std]{lambda^2+{2*a}lambda+{a^2+b^2}}=0$.</p>\n  <p>On solving this equation we get $\\lambda=\\simplify[std]{{-a}+{b}i}$ and $\\lambda=\\simplify[std]{{-a}-{b}i}$.</p>\n  <p>Hence the general solution is:<br/>\\[y = \\simplify[std]{e^({-a}x)(A*sin({b}x)+B*cos({b}x))}\\]<br/>Note that <br/>\\[y'(x)=\\simplify[std]{-{a}e^({-a}x)(A*sin({b}x)+B*cos({b}x))+e^({-a}x)({b}*A*cos({b}x)-{b}*B*sin({b}x))}\\]<br/>Using the conditions $y(0)=1$ and $y'(0)=\\var{c}$ gives:<br/>\\[\\begin{eqnarray*} B &amp;=&amp; 1\\\\ \\simplify[std]{{b}A+{-a}B}&amp;=&amp; \\var{c} \\end{eqnarray*} \\]<br/>This gives $\\displaystyle{A = \\simplify[std]{{c+a}/{b}}}$.</p>\n  <p>Hence the solution is:</p>\n  <p>\\[y=\\simplify[std]{exp({- a} * x) * (cos({b} * x) + &nbsp;({c+a} / {b}) * sin({b} * x))}\\]</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/"}]}