// Numbas version: exam_results_page_options
{"name": "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": [{"variable_groups": [], "variables": {"b1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "b1"}, "c1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c1"}, "a1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a1"}, "d1": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "d1"}}, "ungrouped_variables": ["a1", "c1", "b1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Solve a separable first order ODE, ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"showCorrectAnswer": true, "scripts": {}, "gaps": [{"answer": "{-a1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "scripts": {}, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "partialCredit": 0, "strings": ["."]}, "showpreview": true, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answersimplification": "all", "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "x^2+{2*b1}*x+{(a1+d1)^2-c1^2-2*b1*c1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"message": "<p>Do not enter decimals in your answer, and expand $f(x)$ fully, so that no parentheses appear in the expression.</p>", "showStrings": true, "partialCredit": 0, "strings": [".", "(", ")"]}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "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>", "variableReplacements": [], "marks": 0}], "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 $y=\\alpha\\pm\\sqrt{f(x)}$, where $\\alpha$ is a constant, and $f(x)$ is some function of $x$.</p>", "tags": ["checked2015", "MAS1603", "MAS2105"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>Jan 2016 (WHF)</strong></p>\n<p>The bddy condition determines the solution, so not correct to have $\\pm$ in the solution.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the solution of a first order separable differential equation of the form $(a+y)y'=b+x$.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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</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}\\pm\\simplify{sqrt(x^2+{2*b1}*x+{(a1+d1)^2-c1^2-2*b1*c1})}.\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}