// Numbas version: exam_results_page_options
{"name": "1st Order ODEs - separation 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1st Order ODEs - separation 1", "tags": [], "metadata": {"description": "<p>Separable 1st order ODE with exponentials</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Separation and integration:</p>\n<p>Find the solution of:<br/>\\[\\dfrac{\\text{d}y}{\\text{d}x}={e^\\simplify{x/{A} + {B}*y}}\\]</p>\n<div>Entering formulae: use the syntax c*e^(n*x^p) for $ce^{nx^p}$,<strong> not forgetting the * for multiplying</strong>&nbsp;arbitrary constants.&nbsp;Use&nbsp;<strong>lowercase $c$</strong> for any constant of integration.</div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>The function needs to be split up using \\(e^{a+b}=e^ae^b\\), before it can be separated and integrated:</p>\n<p>$\\displaystyle \\frac{\\text{d}y}{\\text{d}x} = e^\\simplify{ x/{A} + {B} y} = e^\\simplify{x/{A}} e^\\simplify{{B}y}$</p>\n<p>Separating: $\\displaystyle e^\\simplify{{-B} y}\\text{d}y&nbsp; =e^\\simplify{x/{A}}\\text{d}x$</p>\n<p>Integrating both sides: $\\simplify{-{1}/{B} e}^\\simplify{{-B}y} = \\simplify{{A}}e^\\simplify{x/{A}} + C$</p>\n<p>Rearranging:&nbsp;$ e^\\simplify{{-B}y} = \\simplify{-{B}*{A}e}^\\simplify{x/{A}} + c$</p>\n<p>Taking logs:&nbsp;$ {\\var{-B}y} =\\ln\\left(\\simplify{-{B}*{A}e}^\\simplify{x/{A}} + c\\right)$</p>\n<p>Rearranging again: $y=\\simplify{-1/{B}}\\ln\\left(\\simplify{-{B}*{A}e}^\\simplify{x/{A}} + c\\right)$</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"A": {"name": "A", "group": "Ungrouped variables", "definition": "random(-4..4 except 0)", "description": "<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "templateType": "anything"}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "random(-4..4 except 0)", "description": "<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Solution is:</p>\n<p>$y=\\;\\;$[[0]]</p>\n<p>Input all numbers as integers or fractions &ndash; not as decimals.</p>\n<p>The constant of integration should be entered simply as $c$ (ignore muliplying factors).</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "-{1/B}*ln( -({B}*{A}) * e ^ (x/{A}) + c)", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0.5, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input all numbers as integers or fractions.&nbsp;The constant of integration should be entered simply as $c$ (ignore muliplying factors).</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>"}, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}