// Numbas version: exam_results_page_options
{"name": "Shaheen's copy of 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": [{"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"], "variables": {"b": {"name": "b", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..7)", "description": ""}, "s": {"name": "s", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1,-1)", "description": ""}, "a": {"name": "a", "templateType": "anything", "group": "Ungrouped variables", "definition": "s*random(1..5)", "description": ""}, "c": {"name": "c", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..5)", "description": ""}}, "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>", "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>"}, "showQuestionGroupNames": false, "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>", "functions": {}, "parts": [{"scripts": {}, "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, "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"scripts": {}, "answersimplification": "std", "vsetrange": [0, 1], "checkingaccuracy": 1e-05, "vsetrangepoints": 5, "expectedvariablenames": [], "marks": 3, "type": "jme", "checkingtype": "absdiff", "answer": "exp({- a} * x) * (cos({b} * x) + ({c+a} / {b}) * sin({b} * x))", "showpreview": true, "notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "checkvariablenames": false, "showCorrectAnswer": true}]}], "name": "Shaheen's copy of Second order ODE with constant coefficients and boundary conditions", "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "preamble": {"css": "", "js": ""}, "type": "question", "ungrouped_variables": ["a", "s", "b", "c"], "variable_groups": [], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "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/"}]}