// Numbas version: exam_results_page_options
{"name": "MA1002 First order differential equations 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "MA1002 First order differential equations 1", "tags": [], "metadata": {"description": "<p>Solve 4 first order differential equations of two types:$\\displaystyle \\frac{dy}{dx}=\\frac{ax}{y},\\;\\;\\frac{dy}{dx}=\\frac{by}{x},\\;y(2)=1$ for all 4.</p>\n<p>rebelmaths</p>\n<p>&nbsp;</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Separate the variables:</p>\n<p>Find the solutions of the following ordinary differential equations satisfying the condition $y=1$ at $x=2$.</p>\n<p>You may find it instructive to sketch your various solutions (but this is not required for this <span class=\"caps\">CBA</span>).</p>", "advice": "\n  <p>These are all separable first order differential equations.</p>\n  <p>a)</p>\n  <p>$\\displaystyle{\\frac{dy}{dx}=\\frac{y}{\\var{a1}x} \\Rightarrow \\int \\frac{1}{y}\\;dy = \\frac{1}{\\var{a1}}\\int\\frac{1}{x}\\;dx \\Rightarrow \\ln(y)=\\frac{1}{\\var{a1}}\\ln(x)+C}$</p>\n  <p>Exponentiation of both sides then gives $y=Ax^{1/\\var{a1}}$ where we have renamed the constant of integration.</p>\n  <p>To find the particular solution satisfying $y=1$ at $x=2$, we have $\\displaystyle{1=A \\times 2^{1/\\var{a1}} \\Rightarrow A = \\frac{1}{2^{1/\\var{a1}}}}$</p>\n  <p>Hence the solution is $\\displaystyle{y=\\left(\\frac{x}{2}\\right)^{1/\\var{a1}}}$</p>\n  <p>b)</p>\n  <p>$\\displaystyle{\\frac{dy}{dx}=-\\var{a2}\\frac{y}{x} \\Rightarrow \\int \\frac{1}{y}\\;dy = -\\var{a2}\\int\\frac{1}{x}\\;dx \\Rightarrow \\ln(y)=-\\var{a2}\\ln(x)+C}$</p>\n  <p>Exponentiation of both sides then gives $y=Ax^{-\\var{a2}}$ where we have renamed the constant of integration.</p>\n  <p>The particular solution satisfying $y=1$ at $x=2$, gives $A = 2^{\\var{a2}}$</p>\n  <p>Hence the solution is $\\displaystyle{y=\\left(\\frac{2}{x}\\right)^{\\var{a2}}}$</p>\n  <p>c)</p>\n  <p>$\\displaystyle{\\frac{dy}{dx}=\\var{a3}\\frac{x}{y} \\Rightarrow \\int y\\;dy = \\var{a3}\\int x\\;dx \\Rightarrow \\frac{y^2}{2}=\\var{a3}\\frac{x^2}{2}+C\\Rightarrow y^2=\\var{a3}x^2+A}$</p>\n  <p>The particular solution satisfying $y=1$ at $x=2$, gives $A = \\var{1-4*a3}$.</p>\n  <p>Hence the solution is $\\displaystyle{y^2=\\simplify[std]{{a3}x^2+{1-4*a3}}}$.</p>\n  <p>d)</p>\n  <p>$\\displaystyle{\\frac{dy}{dx}=-\\var{a4}\\frac{x}{y} \\Rightarrow \\int y\\;dy = -\\var{a4}\\int x\\;dx \\Rightarrow y^2=-\\var{a4}x^2+A}$</p>\n  <p>The particular solution satisfying $y=1$ at $x=2$, gives $A = \\var{1+4*a4}$.</p>\n  <p>Hence the solution is $\\displaystyle{y^2=\\simplify[std]{{-a4}x^2+{1+4*a4}}}$.</p>\n  ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": [], "variables": {"a2": {"name": "a2", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "a4": {"name": "a4", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything"}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a1", "a3", "a2", "a4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "\n  <p>$\\displaystyle{\\frac{dy}{dx}=\\frac{y}{\\var{a1}x}}$</p>\n  <p>$y=\\;\\;$[[0]]</p>\n  <p>Do not enter decimals in your answer; use only fractions or integers.</p>\n  ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(x/2)^(1/{a1})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input numbers as fractions or integers, not as decimals</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "\n  <p>$\\displaystyle{\\frac{dy}{dx}=-\\var{a2}\\frac{y}{x}}$</p>\n  <p>$y=\\;\\;$[[0]]</p>\n  <p>Do not enter decimals in your answer; use only fractions or integers.</p>\n  ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "(2/x)^({a2})", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [1, 2], "checkVariableNames": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input numbers as fractions or integers, not as decimals</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "\n  <p>$\\displaystyle{\\frac{dy}{dx}=\\var{a3}\\frac{x}{y}}$</p>\n  <p>The solution can be written in the form $y^2=f(x)$. Enter $f(x)$ in the box below</p>\n  <p>$y^2=\\;\\;$[[0]]</p>\n  <p>Do not enter decimals in your answer; use only fractions or integers.</p>\n  ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{1-4*a3} + {a3} * (x ^ 2)", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input numbers as fractions or integers, not as decimals</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "\n  <p>$\\displaystyle{\\frac{dy}{dx}=-\\var{a4}\\frac{x}{y}}$</p>\n  <p>The solution can be written in the form $y^2=g(x)$. Enter $g(x)$ in the box below</p>\n  <p>$y^2=\\;\\;$[[0]]</p>\n  <p>Do not enter decimals in your answer; use only fractions or integers.</p>\n  ", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "answer": "{1+4*a4} + {-a4} * (x ^ 2)", "answerSimplification": "std", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "notallowed": {"strings": ["."], "showStrings": false, "partialCredit": 0, "message": "<p>Input numbers as fractions or integers, not as decimals</p>"}, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}], "contributors": [{"name": "Kieran Mulchrone", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1243/"}]}]}], "contributors": [{"name": "Kieran Mulchrone", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1243/"}]}