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{"name": "Simon's copy of Julie's copy of Second order differential equations 1", "extensions": ["jsxgraph"], "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>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>\n<p>rebelmaths</p>"}, "statement": "<p>Method of undetermined coefficients</p>\n<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>", "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) + ({c+a} / {b}) * sin({b} * x))}\\]</p>", "parts": [{"showCorrectAnswer": true, "variableReplacements": [], "unitTests": [], "scripts": {}, "gaps": [{"showCorrectAnswer": true, "variableReplacements": [], "unitTests": [], "checkingAccuracy": 1e-05, "scripts": {}, "variableReplacementStrategy": "originalfirst", "notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "partialCredit": 0, "strings": ["."], "showStrings": false}, "expectedVariableNames": [], "answer": "exp({- a} * x) * (cos({b} * x) + ({c+a} / {b}) * sin({b} * x))", "marks": 3, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "type": "jme", "checkingType": "absdiff", "vsetRange": [0, 1], "vsetRangePoints": 5, "checkVariableNames": false, "answerSimplification": "std", "failureRate": 1, "showPreview": true}], "sortAnswers": false, "variableReplacementStrategy": "originalfirst", "marks": 0, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "type": "gapfill", "prompt": "\n  <p>Solution is:</p>\n  <p>$y=\\;\\;$[[0]]</p>\n  <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n  "}], "variable_groups": [], "name": "Simon's copy of Julie's copy of Second order differential equations 1", "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "functions": {"graphsolution": {"definition": "var c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//var m=Math.abs((c-d*Math.exp(a))/(-a)*Math.exp(-(1-c*a/(d*Math.exp(a)-c))));\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n  {boundingBox:[-5,30,5,-30],\n   axis:false,\n   showNavigation:true,\n   grid:true});\nvar brd = div.board;  \n  var xas = brd.create('line',[[0,0],[1,0]], { strokeColor: 'black',fixed:true});\n  var xticks = brd.create('ticks',[xas,1],{\ndrawLabels: true,\nlabel: {offset: [-4, -10]},\nminorTicks: 0\n  });\n  var yas = brd.create('line',[[0,0],[0,1]], { strokeColor: 'black',fixed:true});\n  var yticks = brd.create('ticks',[yas,5],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n  }); \nvar p=brd.create('point',[0,1],{fixed:true,name:''});\nvar q=brd.create('line',[[0,0],[1,c]],{fixed:true,name:'',dash:2});\nvar tree;\n//x is the variable in the equation to be input\n  var nscope = new Numbas.jme.Scope([scope,{variables:{x:new Numbas.jme.types.TNum(0)}}]);\n//create a functiongraph from the student input\nfunction userf(t){\nif(tree) {\n  try {\nnscope.variables.x.value = t;\n//the user input is evaluated at x=t\n  var val = Numbas.jme.evaluate(tree,nscope).value;\n  return val;\n  }\n  catch(e) {\nreturn 0;\n  }\n}\nelse\n  return 0;\n}\nvar curve=brd.create('functiongraph',[userf,-10,10],{strokeColor:'red',strokeWidth:2});\n\n   //pick up the student answer and is parsed\n  question.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\nvar expr = question.parts[0].gaps[0].display.studentAnswer();\ntry {\n  tree = Numbas.jme.compile(expr,scope);\n}\ncatch(e) {\n  tree = null;\n}\ncurve.updateCurve();\n\nbrd.update();\n});\n  });  \n\nreturn div;\n  ", "language": "javascript", "parameters": [], "type": "html"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "variables": {"c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..5)", "templateType": "anything", "description": ""}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1..7)", "templateType": "anything", "description": ""}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(-5..5 except 0)", "templateType": "anything", "description": ""}}, "ungrouped_variables": ["a", "c", "b"], "extensions": ["jsxgraph"], "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/"}]}