// Numbas version: exam_results_page_options
{"name": "Simon's copy of Solve a separable first order ODE,", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "prompt": "<p>Solve the equation, and enter the value of $\\alpha$ and the expression for $f(x)$ in the boxes. &nbsp;Do not enter decimals in your answers.</p>\n      <p>$\\alpha=$ [[0]].</p>\n      <p>$f(x)=$ [[1]]. &nbsp;(Expand $f(x)$ fully, so that no parentheses appear in the expression.)</p>", "sortAnswers": false, "type": "gapfill", "unitTests": [], "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "gaps": [{"notallowed": {"partialCredit": 0, "showStrings": false, "strings": ["."], "message": "<p>Do not enter decimals in your answer.</p>"}, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "vsetRange": [0, 1], "answerSimplification": "all", "showCorrectAnswer": true, "checkingType": "absdiff", "type": "jme", "unitTests": [], "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "marks": 1, "failureRate": 1, "expectedVariableNames": [], "answer": "{-a1}", "extendBaseMarkingAlgorithm": true, "checkVariableNames": false}, {"notallowed": {"partialCredit": 0, "showStrings": true, "strings": [".", "(", ")"], "message": "<p>Do not enter decimals in your answer, and expand $f(x)$ fully, so that no parentheses appear in the expression.</p>"}, "variableReplacementStrategy": "originalfirst", "checkingAccuracy": 0.001, "showPreview": true, "vsetRange": [0, 1], "showCorrectAnswer": true, "checkingType": "absdiff", "type": "jme", "unitTests": [], "customMarkingAlgorithm": "", "scripts": {}, "variableReplacements": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "marks": 1, "failureRate": 1, "expectedVariableNames": [], "answer": "x^2+{2*b1}*x+{(a1+d1)^2-c1^2-2*b1*c1}", "extendBaseMarkingAlgorithm": true, "checkVariableNames": false}], "extendBaseMarkingAlgorithm": true}], "tags": [], "name": "Simon's copy of Solve a separable first order ODE,", "preamble": {"css": "", "js": ""}, "variable_groups": [], "extensions": [], "advice": "<p>The differential equation is separable, and can be immediately integrated to give</p>\n<p>\\[\\simplify{{a1}*y+(1/2)*y^2}=\\simplify{{b1}*x+(1/2)*x^2+c},\\]</p>\n<p>or</p>\n<p>\\[\\simplify{(1/2)*(y+{a1})^2-{a1^2}/2}=\\simplify{{b1}*x+(1/2)*x^2+c},\\]</p>\n<p>then the general solution of the equation is</p>\n<p>\\[y=\\var{-a1}\\pm\\simplify{sqrt(x^2+{2*b1}*x+2c+{a1^2})}\\]</p>\n<p>or, upon redefining the constant $c$,</p>\n<p>\\[y=\\var{-a1}\\pm\\simplify{sqrt(x^2+{2*b1}*x+c)}.\\]</p>\n<p>Then we have</p>\n<p>\\[\\var{d1}=y(\\var{c1})=\\var{-a1}\\pm\\simplify[std]{sqrt({c1}^2+{2*b1}*{c1}+c)}=\\var{-a1}\\pm\\simplify{sqrt({c1^2+2*b1*c1}+c)},\\]</p>\n<p>so we require the positive root, and</p>\n<p>\\[c=\\simplify[std]{({a1}+{d1})^2-{c1^2+2*b1*c1}}=\\simplify{{(a1+d1)^2-c1^2-2*b1*c1}}.\\]</p>\n<p>Then the full solution is</p>\n<p>\\[y=\\var{-a1}+\\simplify{sqrt(x^2+{2*b1}*x+{(a1+d1)^2-c1^2-2*b1*c1})}.\\]</p>", "functions": {}, "metadata": {"description": "<p>Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a1", "c1", "b1", "d1"], "variablesTest": {"condition": "d1>-a1", "maxRuns": 100}, "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "variables": {"b1": {"description": "", "definition": "random(1..9)*sign(random(-1,1))", "name": "b1", "group": "Ungrouped variables", "templateType": "anything"}, "a1": {"description": "", "definition": "random(1..9)*sign(random(-1,1))", "name": "a1", "group": "Ungrouped variables", "templateType": "anything"}, "c1": {"description": "", "definition": "random(1..9)*sign(random(-1,1))", "name": "c1", "group": "Ungrouped variables", "templateType": "anything"}, "d1": {"description": "", "definition": "random(1..9)*sign(random(-1,1))", "name": "d1", "group": "Ungrouped variables", "templateType": "anything"}}, "statement": "<p>You are given the differential equation</p>\n<p>\\[(\\var{a1}+y)y'=\\var{b1}+x,\\]</p>\n<p>satisfying $y(\\var{c1})=\\var{d1}$.</p>\n<p>The solution can be written in the form $y=\\alpha+\\sqrt{f(x)}$, where $\\alpha$ is a constant, and $f(x)$ is some function of $x$.</p>", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}