// Numbas version: finer_feedback_settings
{"name": "Simon's copy of Julie's copy of Second order differential equations 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the general solution of $y''+2py'+(p^2-q^2)y=x$ in the form &nbsp;$Ae^{ax}+Be^{bx}+y_{PI}(x),\\;y_{PI}(x)$ a particular integral.</p>\n<p>rebelmaths</p>"}, "statement": "\n    <p>Find the complete general solution of the equation:<br/>\\[\\simplify[std]{d^2y/dx^2+{2*p}*(dy/dx)+{p^2-q^2}y=x}\\]<br/>in the form $\\displaystyle{Ae^{ax}+Be^{bx}+y_{PI}(x)}$ where $A$ and $B$ are arbitrary constants and $y_{PI}$ is a particular integral.</p>\n  <p>Note that we require that $a \\gt b$.</p>\n    ", "showQuestionGroupNames": false, "ungrouped_variables": ["q1", "p", "q"], "parts": [{"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "showCorrectAnswer": true, "minValue": "{q-p}", "allowFractions": false, "scripts": {}, "maxValue": "{q-p}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showPrecisionHint": false, "marks": 1, "type": "numberentry"}, {"variableReplacements": [], "showCorrectAnswer": true, "minValue": "{-q-p}", "allowFractions": false, "scripts": {}, "maxValue": "{-q-p}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showPrecisionHint": false, "marks": 1, "type": "numberentry"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "prompt": "\n      \n      \n      <p>$a=\\;\\;$[[0]]$\\;\\;\\;b=\\;\\;$[[1]].  Remember that we require $a \\gt b$</p>\n      \n      \n      \n        "}, {"variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"showCorrectAnswer": true, "vsetrangepoints": 5, "showpreview": true, "variableReplacementStrategy": "originalfirst", "notallowed": {"message": "<p>Input all numbers as fractions or integers and not as decimals.</p>", "strings": ["."], "showStrings": false, "partialCredit": 0}, "scripts": {}, "expectedvariablenames": [], "marks": 3, "variableReplacements": [], "answersimplification": "std", "type": "jme", "answer": "x/{p^2-q^2}-{2*p}/{(p^2-q^2)^2}", "checkingaccuracy": 1e-05, "checkvariablenames": false, "checkingtype": "reldiff", "vsetrange": [0, 1]}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 0, "prompt": "\n      \n      \n      <p>$\\displaystyle{y_{PI}=\\;\\;}$[[0]].</p>\n      \n      \n      \n      <p>Input all numbers as fractions or integers and not as decimals.</p>\n      \n      \n      \n        "}], "variable_groups": [], "name": "Simon's copy of Julie's copy of Second order differential equations 5", "extensions": [], "tags": ["2nd order differential equation", "Calculus", "calculus", "CF", "complementary function", "constant coefficients", "Differential equations", "differential equations", "general solution", "linear differential equation", "ODE", "ode", "particular integral", "PI", "rebelmaths", "second order differential equation", "solving differential equations"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"css": "", "js": ""}, "variables": {"q": {"definition": "if(q1=abs(p),q1+1,q1)", "templateType": "anything", "name": "q", "group": "Ungrouped variables", "description": ""}, "q1": {"definition": "random(1..8)", "templateType": "anything", "name": "q1", "group": "Ungrouped variables", "description": ""}, "p": {"definition": "random(-6..6)", "templateType": "anything", "name": "p", "group": "Ungrouped variables", "description": ""}}, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "advice": "\n    <p>First we find the Complementary Function (CF) i.e. the solution of:</p>\n  <p>\\[\\simplify[std]{d^2y/dx^2+{2*p}*(dy/dx)+{p^2-q^2}y=0}\\]</p>\n  <p>The auxillary equation is $\\simplify[std]{lambda^2+{2*p}lambda+{p^2-q^2}=0} \\Rightarrow \\simplify[std]{(lambda+{p-q})(lambda+{p+q})=0}\\Rightarrow \\lambda = \\var{q-p}\\textrm{ or }\\lambda = \\var{-q-p} $</p>\n  <p>Hence the CF is $\\displaystyle{y_{CF}(x)=\\simplify[std]{A*e^({q-p}x)+ B*e^({-q-p}x)}}$ for $A$, $B$ arbitrary constants.</p>\n  <p>So $a=\\var{q-p}$ and $b=\\var{-q-p}$ as we required $a \\gt b$.</p>\n  <p>To find the Particular Integral (PI) for $\\displaystyle{\\simplify[std]{d^2y/dx^2+{2*p}*(dy/dx)+{p^2-q^2}y=x}}$, we observe that</p>\n  <p>$y_{PI}(x)=cx+d$ is a possible PI for constants $c,\\;d$ to be found.</p>\n  <p>Substituting $y_{PI}(x)=cx+d$ in to the equation gives:</p>\n  <p>$\\displaystyle \\simplify[std]{{2*p}c+{p^2-q^2}*(cx+d)} =x \\Rightarrow \\simplify[std]{{p^2-q^2}cx+{p^2-q^2}d+{2*p}c}= x \\Rightarrow c=\\simplify[std]{1/{p^2-q^2}},\\;\\;d = \\simplify[std]{{-2p}/{(p^2-q^2)^2}}$</p>\n  <p>So the particular integral is $\\displaystyle{\\simplify[std]{x/{p^2-q^2}-{2*p}/{(p^2-q^2)^2}}}$</p>\n    ", "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}