// Numbas version: exam_results_page_options
{"name": "1st order ODE - Initial Conditions and new value", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1st order ODE - Initial Conditions and new value", "tags": [], "metadata": {"description": "<p>Given an ODE solution: $y=&nbsp; b x^n + cx$</p><p>Use a condition: $y(1)=a$ to find the particular solution and hence the value of $y(d)$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Starting with&nbsp;the General&nbsp;solution of&nbsp;\\(\\frac{dy}{dx}-\\frac{y}{x}=\\simplify[std]{{b}x^{n}}\\), which is given as:</p>\n<p>\\[y=&nbsp; \\simplify{{b}/{n}x^{n+1}}+cx\\]</p>\n<p>Use the condition:&nbsp;$\\displaystyle{y(1)=\\simplify[std]{{b}/{n-1}}}$ to find the particular solution and hence the value of \\(y(\\var{x1})\\).</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>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "advice": "<p>Starting with the General Solution:&nbsp;</p>\n<p>\\[y=&nbsp; \\simplify{{b}/{n}x^{n+1}}+cx\\]<br/>to determine $c$ use the condition $\\displaystyle{y(1)=\\simplify[std]{{b}/{n-1}}}$ to obtain:</p>\n<p>\\[\\simplify[std]{{b}/{n-1}}=\\simplify[std]{{b}/{n}+c}\\]<br/>\\[c = \\simplify[std]{{b}/{n-1}} - \\simplify{{b}/{n}} =\\simplify{{b}/{n-1} - {b}/{n}}\\]<br/>and so the particular solution is:\\[y=&nbsp; \\simplify{{b}/{n}x^{n+1}+({b}/{n-1} - {b}/{n}) * x}\\]</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\">Putting in the value \\( x= \\var{x1} \\) we obtain\\[y=\\simplify{{b}/{n}*{x1}^{n+1}+({b}/{n-1} - {b}/{n})*{x1}}\\]</div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"b": {"name": "b", "group": "Ungrouped variables", "definition": "n*(n-1)", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "({b}/{n})*{x1}^({n}+1) + ({b}/({n}-1) - {b}/{n})*{x1}", "description": "<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "b", "x1", "answer"], "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<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>\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": "{answer}", "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.</p>"}, "valuegenerators": []}], "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/"}, {"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}]}]}], "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/"}, {"name": "Andrew Barnes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3725/"}]}