// Numbas version: exam_results_page_options {"name": "First Order Differential Equation 2 [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 2 [Explore]", "tags": [], "metadata": {"description": "
First order integral.
", "licence": "All rights reserved"}, "statement": "The following first order differential equation is given.
\n\n$\\frac{dy}{dx} = \\frac{\\var{a}}{{\\var{b}}x+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$
", "advice": "$\\frac{dy}{dx} = \\frac{\\var{a}}{{\\var{b}}x+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$
\n\\[ \\int{\\frac{\\var{a}}{{\\var{b}}x+{\\var{c}}}dx} \\]
\nMove constant outside
\n\\[ \\var{a}\\int{\\frac{1}{{\\var{b}}x+{\\var{c}}}.dx} \\]
\nlet \\[ u =\\var{b}x+\\var{c}\\]
\n\\[\\frac{du}{dx}=\\var{b} \\]
\n\\[\\therefore dx=\\frac{du}{\\var{b}}\\]
\nSub back into equation
\n\\[ \\var{a}\\int{\\frac{1}{u}.\\frac{du}{\\var{b}}} \\]
\nThe two x values cancel, leaving
\n\\[ \\frac{\\var{a}}{\\var{b}}\\int{\\frac{1}{u}.du}=\\frac{\\var{a}}{\\var{b}} ln(u)+C \\]
\nSub back in the original denominator
\n\\[y= \\frac{\\var{a}}{\\var{b}} ln(\\var{b}x+\\var{c})+C \\]
", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(1 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2 .. 8#1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1 .. 6#1)", "description": "", "templateType": "randrange", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(1 .. 5#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "Cint": {"name": "Cint", "group": "Ungrouped variables", "definition": "precround({d}-({a}/{b})ln(({b}*0)+{c}),2)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "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": "Find the general solution for the expression.
\nEnter values as integers or fractions.
\n$y=$[[0]]
", "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}/{b})ln({b}*x+{c})+C", "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": "The general solution for part a can be expressed as,
\n$y=\\frac{\\var{a}}{\\var{b}}ln(\\var{b}x+\\var{c})+C$
\nEvaluate the constant of integration, C. Input values decimals.
\n$C=$[[0]]
", "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": "If the constant of integration can be expressed as
\n\\[ C =\\var{Cint} \\]
\nstate the particular solution
\n$y=$[[0]]
", "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}/{b})ln({b}*x+{c})+{Cint}", "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/"}]}