// Numbas version: exam_results_page_options
{"name": "First Order Differential Equation 1 [Explore]", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "First Order Differential Equation 1 [Explore]", "tags": [], "metadata": {"description": "<p>Separable equation for integration.</p>", "licence": "All rights reserved"}, "statement": "<p>The following first order differential equation is given.</p>\n<p></p>\n<p>$\\frac{dy}{dx} = xy^\\var{a}$, where $y=\\var{b}$ when $x=0$</p>", "advice": "<p>$\\frac{dy}{dx} = xy^\\var{a}$, where $y=\\var{b}$ when $x=0$</p>\n<p>Separate the variables</p>\n<p>\\[ y^\\var{-a}.dy = x.dx \\]</p>\n<p>Integrate each side</p>\n<p>\\[ \\int{y^\\var{-a}.dy} = \\int{x}.dx \\]</p>\n<p>\\[\\frac{y^\\var{-a+1}}{\\var{-a+1}}=\\frac{x^2}{2}+C \\] ...general solution</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2 .. 4#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(1 .. 12#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "Cint": {"name": "Cint", "group": "Ungrouped variables", "definition": "precround({b}^(-{a}+1)/(-{a}+1),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "Cint"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "General Solution", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Constant of Integration", "rawLabel": "", "otherPart": 1, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": true}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "General Solution", "prompt": "<p>Find the general solution for the expression.</p>\n<p>$\\frac{dy}{dx} = xy^\\var{a}$</p>\n<p>Enter values as integers or fractions. Use C as the constant of integration.</p>\n<p>$y =$&nbsp; [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 0.", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "((-{a}+1)*(x^2/2 + C))^(1/(-{a}+1))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Constant of Integration", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [{"label": "Particular Solution", "rawLabel": "", "otherPart": 2, "variableReplacements": [], "availabilityCondition": "answered", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": true}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Constant of Integration", "prompt": "<p>The general solution for part a can be expressed as,</p>\n<p>\\[y =( {\\var{-a+1}}(\\frac{x^{2}}{2} + C))^{\\frac{1}{\\var{-a+1}}}\\]</p>\n<p>Evaluate the constant of integration, C, for the initial conditions $y=\\var{b}$ when $x=0$. Input values as decimal.</p>\n<p>$C=$ [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 1.", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.95*{Cint}", "maxValue": "1.05*{Cint}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": true, "customName": "Particular Solution", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": "Particular Solution", "prompt": "<p>For a constant of integration&nbsp;</p>\n<p>\\[C=\\frac{\\var{b}^{\\var{-a+1}}}{\\var{-a+1}}=\\var{Cint}\\]</p>\n<p>Sub this back into the general solution and write the particular solution in terms of y.</p>\n<p>$y =$ [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 2.", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({-a+1}*((x^2)/2 +{Cint}))^((1)/({-a+1}))", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "explore", "maxMarks": "8", "objectives": [{"name": "General Solution", "limit": "4"}, {"name": "Constant of Integration", "limit": "2"}, {"name": "Particular Solution", "limit": "2"}], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}]}], "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}