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{"name": "Differential Equations: Separation of Variables 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differential Equations: Separation of Variables 3", "tags": [], "metadata": {"description": "<p>Solving a differential equation of the form $\\frac{dy}{dx}=\\frac{ax^n}{y}$ using separation of variables.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Solve the differential equation \\[ \\frac{dy}{dx}=\\frac{\\var{a}x^\\var{n}}{y}.\\]</p>", "advice": "<p>If we have a differential equation of the form \\[ \\frac{dy}{dx} = f(x) g(y),\\] we are able to find a solution to this equation using the method Separation of Variables.</p>\n<p>We can rewrite the above equation in the form \\[ \\frac{1}{g(y)} \\frac{dy}{dx} = f(x).\\]</p>\n<p>If we then integrate with respect to $x$:&nbsp;</p>\n<p>\\[ \\begin{split} &amp;\\int \\frac{1}{g(y)} &amp;\\frac{dy}{dx} dx = \\int f(x) dx, \\\\\\\\ \\implies &amp;\\int \\frac{1}{g(y)} &amp;dy = \\int f(x) dx.&nbsp; \\end{split} \\]</p>\n<p>Integrating both sides, we can write $y$ as a function of $x$.&nbsp;</p>\n<p>Following this method for $\\frac{dy}{dx}=\\frac{\\var{a}x^\\var{n}}{y}$:</p>\n<p>\\[\\frac{dy}{dx}=\\frac{\\var{a}x^\\var{n}}{y}\\]</p>\n<p>\\[ \\begin{split} &amp;\\frac{dy}{dx}=\\frac{\\var{a}x^\\var{n}}{y} \\\\ \\implies&nbsp; y\\,&amp; \\frac{dy}{dx} = \\simplify{{a}x^{n}}. \\end{split} \\]</p>\n<p>Integrating both sides with respect to $x$:</p>\n<p>\\[ \\begin{split}&nbsp; \\int&nbsp; y \\, \\frac{dy}{dx}\\, dx &amp;= \\int \\simplify{{a}x^{n}} \\, dx, \\\\ \\\\ \\implies \\int y \\,dy &amp;= \\int\\simplify{{a}x^{n}} \\, dx.&nbsp; &nbsp;\\end{split} \\]</p>\n<p>Taking&nbsp;the integral of both sides, we are able to find a solution to the differential equation:</p>\n<p>\\[ \\begin{split} \\int y \\,dy &amp;=\\simplify{{a}x^{n}} \\, dx \\\\ \\frac{y^2}{2}&amp;\\,= \\simplify{{a}x^{n+1}/{n+1} +c}, \\\\ y^2 &amp;\\,= \\simplify{{2a}x^{n+1}/{n+1} + 2c} \\\\&nbsp; y &amp;\\,= \\pm\\sqrt{\\simplify{{2a}x^{n+1}/{n+1} + 2c}}.&nbsp; \\end{split} \\]</p>\n<p><em>&nbsp;</em></p>\n<p><em><strong>Note</strong>: You can also have the answer&nbsp;</em></p>\n<p><em>\\[ y = \\pm \\sqrt{\\simplify{{2a}x^{n+1}/{n+1}+D}}, \\]</em></p>\n<p><em>where $D=2c$. This shows there is still a constant, but indicates it is different to the original constant of integration, $c$.</em></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "n"], "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>$y=\\,$[[0]]</p>", "gaps": [{"type": "jme", "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, "alternatives": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "", "useAlternativeFeedback": false, "answer": "sqrt({2*a/(n+1)}x^{n+1}+c)", "answerSimplification": "basic", "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": ""}]}], "answer": "sqrt({2*a/(n+1)}x^{n+1}+2c)", "answerSimplification": "basic", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}