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{"name": "Separable order ODE 2 violeta", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"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>", "marks": 0, "scripts": {}, "type": "gapfill", "variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showpreview": true, "scripts": {}, "expectedvariablenames": [], "type": "jme", "variableReplacements": [], "notallowed": {"partialCredit": 0, "showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."]}, "answersimplification": "all", "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answer": "{-a1}", "vsetrange": [0, 1], "showCorrectAnswer": true}, {"variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showpreview": true, "scripts": {}, "expectedvariablenames": [], "type": "jme", "variableReplacements": [], "notallowed": {"partialCredit": 0, "showStrings": true, "message": "<p>Do not enter decimals in your answer, and expand $f(x)$ fully, so that no parentheses appear in the expression.</p>", "strings": [".", "(", ")"]}, "vsetrangepoints": 5, "marks": 1, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "answer": "x^2+{2*b1}*x+{(a1+d1)^2-c1^2-2*b1*c1}", "vsetrange": [0, 1], "showCorrectAnswer": true}], "showCorrectAnswer": true}], "tags": ["checked2015", "MAS1603", "MAS2105"], "variable_groups": [], "metadata": {"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>", "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>"}, "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>", "preamble": {"css": "", "js": ""}, "functions": {}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered"}], "type": "question", "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>", "variables": {"c1": {"definition": "random(1..9)*sign(random(-1,1))", "description": "", "group": "Ungrouped variables", "name": "c1", "templateType": "anything"}, "b1": {"definition": "random(1..9)*sign(random(-1,1))", "description": "", "group": "Ungrouped variables", "name": "b1", "templateType": "anything"}, "a1": {"definition": "random(1..9)*sign(random(-1,1))", "description": "", "group": "Ungrouped variables", "name": "a1", "templateType": "anything"}, "d1": {"definition": "random(1..9)*sign(random(-1,1))", "description": "", "group": "Ungrouped variables", "name": "d1", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "name": "Separable order ODE 2 violeta", "ungrouped_variables": ["a1", "c1", "b1", "d1"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Violeta CIT", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1030/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Violeta CIT", 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