// Numbas version: finer_feedback_settings
{"name": "1 ODE Eq 1 Explore", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1 ODE Eq 1 Explore", "tags": [], "metadata": {"description": "<p>Solving first order differentials by separation of variables.</p>", "licence": "All rights reserved"}, "statement": "<p>The following first order differential equation is given.</p>\n<p></p>\n<p>$\\frac{dy}{dx} = xy$, where $y=\\var{b}$ when $x=0$</p>", "advice": "<p>The first step in this equation is to separate the variables to each side of the expression</p>\n<p>$\\frac{dy}{dx} = xy$</p>\n<p>Separate the differentials, multiplying across by $dx$</p>\n<p>$dy = xy.dx$</p>\n<p>Move any $y$ terms over to the differentials $dy$</p>\n<p>$\\frac{dy}{y} = x.dx$</p>\n<p></p>\n<p>Each side can now be integrated</p>\n<p>\\[ln(y)=\\frac{x^{2}}{2}+C\\]&nbsp; ...general solution</p>\n<p>Rewrite in terms of y</p>\n<p>\\[ y = e^{\\frac{x^{2}}{2}+C} \\]</p>\n<p>To evaluate C, input the known values of y and x&nbsp; $y=\\var{b}$ when $x=0$.</p>\n<p></p>\n<p>\\[ln(\\var{b})= \\frac{(0)^2}{2}+C\\]</p>\n<p><br/>\\[C= ln(\\var{b})=\\var{Cint}\\]</p>\n<p></p>\n<p>State the particular solution in respect of y&nbsp;</p>\n<p>\\[ y = e^{\\frac{x^{2}}{2}+\\var{Cint}}\\]</p>\n<p>An alternative method of stating this is</p>\n<p>\\[ y = e^{\\var{Cint}}.e^{\\frac{x^{2}}{2}}=\\var{e^{Cint}}e^{\\frac{x^{2}}{2}}\\]</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(2 .. 8#0.5)", "description": "", "templateType": "randrange", "can_override": false}, "Cint": {"name": "Cint", "group": "Ungrouped variables", "definition": "precround(ln({b}),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": "", "penalty": "", "penaltyAmount": 0, "showPenaltyHint": true, "lockAfterLeaving": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the general solution for the expression.</p>\n<p>$\\frac{dy}{dx} = xy$</p>\n<p>Enter values as integers or fractions. Use C as the constant of integration.</p>\n<p>y=[[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 2.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "e^(x^2/2 + 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": false}], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The general solution can be expressed as,</p>\n<p>\\[y= e^{\\frac{x^2}{2}+C }\\]</p>\n<p>Evaluate the constant of integration, C, for the initial conditions $y=\\var{b}$ when $x=0$. Input values as integers or fractions.</p>\n<p>C=[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "Gap 0.", "marks": 1, "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": true, "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": null, "prompt": "<p>For a constant of integration&nbsp;</p>\n<p>\\[C=ln(\\var{b})=\\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=&nbsp; [[0]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Gap 1.", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "e({Cint})*e^(x^2/2)", "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": ""}]}, {"type": "jme", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "{e^(Cint)}e^(x^2/2)", "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": ""}]}], "answer": "e^(x^2/2+{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": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}], "resources": []}]}], "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}]}