// Numbas version: exam_results_page_options
{"name": "Section 4 (practice set)", "metadata": {"description": "", "licence": "None specified"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "1st order ODE - Initial Conditions and new value", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "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"]}, "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"}, {"name": "1st Order ODEs - separation 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "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"]}, "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"}, {"name": "Integrating Factor 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "tags": [], "metadata": {"description": "<p>Using the IF to find the General Solution.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Integrating Factor Method:</p>\n<p>Find the General&nbsp;Solution of:<br/>\\[\\simplify{{A}x^{{m}}}\\simplify{(d y)/(d x)+{B}}x^\\simplify{{n}+{m}}y=\\simplify{{D}x}^\\simplify{{n}+{m}}\\]</p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"><span style=\"text-decoration: underline;\"><strong>Enter your answer as a formula:</strong></span> use the syntax c*e^(n*x^p) for $ce^{nx^p}$,<strong>&nbsp;not forgetting the * for multiplying</strong>&nbsp;arbitrary constants.&nbsp;Use&nbsp;<strong>lowercase $c$</strong>&nbsp;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>\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>\\(\\simplify{{A}x^{{m}}}\\simplify{(d y)/(d x)+{B}}x^\\simplify{{n}+{m}}y=\\simplify{{D}x}^\\simplify{{n}+{m}}\\) can be solved by the integrating factor method.</p>\n<p>Firstly divide both sides by \\(\\simplify{{A}x^{{m}}}\\) to obtain \\(\\simplify{(d y)/(d x)+{B}/{A}}\\simplify{x^{n}}y=\\simplify{{D}/{A}*x^{n}}\\)</p>\n<p>This is now in the standard form $\\displaystyle{\\frac{dy}{dx}+f(x)y=r(x)}$ which can be solved&nbsp;using the integrating factor $I=e^{\\int f(x)dx}$</p>\n<p></p>\n<p>In&nbsp;this example, $f(x) =\\simplify{{B}/{A} x^{n}}$ so&nbsp;$e^{\\int f(x)dx}=e^{\\int \\simplify{{B}/{A} x^{n} d x}}=e^{\\simplify{{B}/{(n+1)*A} x^{n+1}}}$</p>\n<p></p>\n<p>The integrating factor formula \\( y = \\frac{1}{I}\\int r(x) I dx\\) can be used with \\( r(x) = \\simplify{{D}/{A}x^{n}} \\) to obtain:</p>\n<p>\\[\\displaystyle&nbsp; y =e^{\\simplify{-{B}/{(n+1)*A} x^{n+1}}} \\int&nbsp;\\simplify{{D}/{A}x^{{n}}}e^{\\simplify{{B}/{({n+1})*A} x^{n+1}}} dx\\]</p>\n<p>This can be solved using the substitution: \\( u= \\simplify{{B}/{(n+1)*A} x^{n+1}}\\) so \\( \\frac{du}{dx} =&nbsp;\\simplify{{B}/{A} x^{n}}\\) and therefore&nbsp;\\(\\simplify{x^{n}} dx =&nbsp;\\simplify{{A}/{B}}du\\), which gives:</p>\n<p>\\[\\displaystyle&nbsp; y = e^{-u} \\int \\simplify{{D}/{B}e^u} du&nbsp; = e^{-u} \\left[\\simplify{{D}/{B} e^u} + c\\right] =&nbsp; \\simplify{{D}/{B}} + ce^{-u}\\]&nbsp;</p>\n<p>Substituting back \\( u= \\simplify{{B}/{(n+1)*A} x^{n+1}}\\) results&nbsp;in a general solution: \\( y =&nbsp;&nbsp;\\simplify{{D}/{B}} + ce^{\\simplify{-{B}/{(n+1)*A} x^{n+1}}} \\)&nbsp;</p>\n<p></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"]}, "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "1", "description": "", "templateType": "anything"}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "A*(n+1)*H", "description": "<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "templateType": "anything"}, "H": {"name": "H", "group": "Ungrouped variables", "definition": "random(-1..3 except 0)", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "G": {"name": "G", "group": "Ungrouped variables", "definition": "random(-1..3 except 0)", "description": "", "templateType": "anything"}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "G*B", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["G", "H", "A", "n", "B", "D", "m", "c"], "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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{G} + c * e^(-{H}*x^{n+1})", "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": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Integrating Factor 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "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/"}], "tags": [], "metadata": {"description": "<p>Find the solution of&nbsp;$\\displaystyle x\\frac{dy}{dx}+ay=bx^n,\\;\\;y(1)=c$</p>\n<p></p>\n<div id=\"vidyowebrtcscreenshare_is_installed\"></div>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Integrating Factor Method:</p>\n<p>Find the solution of:<br/>\\[x\\frac{dy}{dx}+\\var{a}y=\\simplify[std]{{b}x^{n}}\\]</p>\n<p>which satisfies $\\displaystyle{y(1)=\\simplify[std]{{b*(c+1)}/{a+n}}}$</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>", "advice": "<p>$\\displaystyle{x\\frac{dy}{dx}+\\var{a}y=\\simplify[std]{{b}x^{n}}}$ can be solved by the integrating factor method.</p>\n<p>Firstly divide both sides by $x$ to obtain&nbsp;$\\displaystyle{\\frac{dy}{dx}+\\frac{\\var{a}}{x}y=\\simplify[std]{{b}x^{n-1}}}$</p>\n<p>This is now in the standard form $\\displaystyle{\\frac{dy}{dx}+f(x)y=r(x)}$ which can be solved&nbsp;using the integrating factor $I = e^{\\int f(x)dx}$</p>\n<p></p>\n<p>In&nbsp;this example, $f(x) =&nbsp;\\frac{\\var{a}}{x}$ so&nbsp;$I=e^{\\int f(x)dx}=e^{\\int{\\frac{\\var{a}}{x}dx}}=e^{\\var{a}\\ln(x)}=x^\\var{a}$</p>\n<p>Using the integrating factor formula \\( y = \\frac{1}{I} \\int r(x) I dx\\) with \\(r(x) =\\simplify[std]{{b}x^{n-1}}\\) gives a solution:<br/>\\[y=x^{\\var{-a}} \\left[ \\simplify[std]{{b}/{a+n}x^{a+n}+c}\\right] \\]<br/>to determine $c$ use the condition $\\displaystyle{y(1)=\\simplify[std]{{b*(c+1)}/{a+n}}}$ to obtain:</p>\n<p>\\[\\simplify[std]{{b*(c+1)}/{a+n}}=\\simplify[std]{{b}/{a+n}+c}\\]<br/>\\[c = \\simplify[std]{{b*(c+1)}/{a+n}} - \\simplify[std]{{b}/{a+n} = {b*c}/{a+n}}\\]<br/>and so the solution is:<br/>\\[y=x^{\\var{-a}}\\left[\\simplify[std]{{b}/{a+n}x^{a+n}+{b*c}/{a+n}}\\right] \\Rightarrow y=\\simplify[std]{{b}/{a+n}*(x^{n}+{c}*x^{-a})}\\]</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", "!noLeadingMinus", "!collectNumbers", "fractionNumbers"]}, "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything"}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "random(1..3)", "description": "", "templateType": "anything"}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "<p></p>\n<div 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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>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "3", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "({b} / {(a + n)}) * (x ^ {n} + ({c} * (x ^ { - a})))", "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, 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true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Nick McCullen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/953/"}], "extensions": [], "custom_part_types": [], "resources": []}