// Numbas version: finer_feedback_settings
{"name": "Julie's copy of First order differential equations 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "c", "b", "n"], "name": "Julie's copy of First order differential equations 5", "tags": ["1st order differential equation", "boundary condition", "boundary conditions", "Calculus", "calculus", "Differential equations", "differential equations", "first order differential equation", "integrating factor", "integrating factor method", "linear first order differential equation", "ODE", "ode", "solving differential equations", "steps", "Steps"], "advice": "\n\n\n<p>$\\displaystyle{x\\frac{dy}{dx}+\\var{a}y=\\simplify[std]{{b}x^{n}}}$ is a linear equation of the special type where we can multiply both sides by $\\simplify[std]{x^{a-1}}$ to get:<br/>\\[\\frac{d(x^{\\var{a}}y)}{dx}=\\simplify[std]{{b}x^{a+n-1}}\\]<br/>We can integrate both sides to get:<br/>\\[x^{\\var{a}}y=\\simplify[std]{{b}/{a+n}x^{a+n}+A}\\]<br/>to determine $A$ we use the condition $\\displaystyle{y(1)=\\simplify[std]{{b*(c+1)}/{a+n}}}$ and we see that:<br/>\\[A = \\simplify[std]{{b*(c+1)}/{a+n} - {b}/{a+n} = {b*c}/{a+n}}\\]<br/>and so the solution is:<br/>\\[x^{\\var{a}}y=\\simplify[std]{{b}/{a+n}x^{a+n}+{b*c}/{a+n}} \\Rightarrow y=\\simplify[std]{{b}/{a+n}*(x^{n}+{c}*x^{-a})}\\]</p>\n\n\n", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n<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>You can get help by clicking on Steps &ndash; you will not lose any marks by doing so.</p>\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Input all numbers as integers or fractions.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacements": [], "expectedvariablenames": [], "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "vsetrangepoints": 5, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "std", "scripts": {}, "answer": "({b} / {(a + n)}) * (x ^ {n} + ({c} * (x ^ { - a})))", "marks": 3, "checkvariablenames": false, "checkingtype": "absdiff", "vsetrange": [0.5, 1]}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "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>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"a": {"definition": "random(2..8)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(1..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "n": {"definition": "random(1..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}}, "metadata": {"description": "<p>Find the solution of&nbsp;$\\displaystyle x\\frac{dy}{dx}+ay=bx^n,\\;\\;y(1)=c$</p>\n<p>rebelmaths</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}]}