// Numbas version: exam_results_page_options
{"name": "Simon's copy of Julie's copy of First order differential equations 4", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Find the solution of $\\displaystyle&nbsp;\\frac{dy}{dx}=\\frac{1+y^2}{a+bx}$&nbsp;which satisfies $y(1)=c$</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p><strong>Separation of variables</strong></p>\n<p>Use separation of variables to solve the following differential equation:</p>\n<p>\\[\\frac{dy}{dx}=\\simplify[std]{(1+y^2)/({a}+{b}x)}\\]<br/>which satisfies $y(1)=\\var{u}$</p>\n<p>Note that if $\\pi$ is in your expression you enter it as pi.</p>", "variable_groups": [], "extensions": [], "parts": [{"scripts": {}, "variableReplacements": [], "marks": 0, "type": "gapfill", "gaps": [{"answerSimplification": "std", "checkVariableNames": false, "answer": "tan(({((t * (t -1)) / 2)} * (pi / 3)) + ({((t -1) * (t -2)) / 2} * (pi / 4)) + ((1 / {b}) * ln(abs(({a} + ({b} * x)) / {a + b}))))", "scripts": {}, "variableReplacements": [], "marks": 3, "checkingAccuracy": 1e-06, "notallowed": {"strings": ["."], "message": "<p>Input all numbers as integers or fractions.</p>", "partialCredit": 0, "showStrings": false}, "type": "jme", "failureRate": 1, "expectedVariableNames": [], "vsetRange": [0, 1], "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true, "vsetRangePoints": 5, "variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "showPreview": true, "customMarkingAlgorithm": "", "showFeedbackIcon": true}], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "unitTests": [], "showCorrectAnswer": true, "sortAnswers": false, "prompt": "\n  <p>Solution is:</p>\n  <p><br/>$y=\\;\\;$[[0]]</p>\n  <p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n  ", "showFeedbackIcon": true}], "tags": [], "name": "Simon's copy of Julie's copy of First order differential equations 4", "ungrouped_variables": ["a", "b", "u", "t", "v"], "functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "preamble": {"js": "", "css": ""}, "variables": {"t": {"description": "", "templateType": "anything", "name": "t", "definition": "random(0,1,2)", "group": "Ungrouped variables"}, "v": {"description": "", "templateType": "anything", "name": "v", "definition": "switch(t=0,'$\\\\pi/4$',t=1,0,'$\\\\pi/3$')", "group": "Ungrouped variables"}, "u": {"description": "", "templateType": "anything", "name": "u", "definition": "switch(t=0,1,t=1,0,'$\\\\sqrt{3}$')", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "anything", "name": "b", "definition": "random(2..9)", "group": "Ungrouped variables"}, "a": {"description": "", "templateType": "anything", "name": "a", "definition": "random(1..9)", "group": "Ungrouped variables"}}, "advice": "\n  <p>We can separate the variables to get</p>\n  <p>\\[\\frac{dy}{dx}=&nbsp;\\simplify[std]{(1+y^2)/({a}+{b}x)}&nbsp;\\Rightarrow \\frac{1}{1+y^2}\\frac{dy}{dx}=\\simplify[std]{{a}+{b}x}\\]</p>\n  <p>On integrating we get:<br/>\\[\\arctan(y)=\\frac{1}{\\var{b}}\\ln\\left(\\left|\\var{a}+\\var{b}x\\right|\\right)+A \\Rightarrow y=\\tan\\left(\\frac{1}{\\var{b}}\\ln\\left(\\left|\\var{a}+\\var{b}x\\right|\\right)+A\\right)\\]<br/>To fix the arbitrary constant of integration we use the condition $y(1)=\\var{u}$.</p>\n  <p>As $\\arctan(\\var{u})=\\var{v}$ we see that $\\displaystyle{A = \\var{v}-\\frac{1}{\\var{b}}\\ln(|\\var{a+b}|)}$.</p>\n  <p>Hence the solution is</p>\n  <p>\\[y=\\simplify[std]{tan(({((t * (t -1)) / 2)} * (pi / 3)) + ({((t -1) * (t -2)) / 2} * (pi / 4)) + ((1 / {b}) * ln(abs(({a} + ({b} * x)) / {a + b}))))}\\]</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "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/"}]}