// Numbas version: finer_feedback_settings
{"name": "Solve a separable first order ODE with trig functions, ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"answer": "{d1}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{beta}", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "answersimplification": "fractionnumbers", "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}, {"answer": "{if(is_cosec,1,0)}*cosec(x)+{if(is_cosec,0,1)}*sin(x)", "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "notallowed": {"showStrings": false, "message": "<p>Do not enter decimals in your answer.</p>", "strings": ["."], "partialCredit": 0}, "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 1, "vsetrangepoints": 5}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Solve the equation, and enter the values of $\\alpha$ and $\\beta$, and the expression for $f(x)$ in the boxes. &nbsp;Do not enter decimals in your answers.</p>\n<p>$\\alpha=$ [[0]]</p>\n<p>$\\beta=$ [[1]]</p>\n<p>$f(x)=$ [[2]]</p>", "variableReplacements": [], "marks": 0}], "variables": {"beta": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs({b1}/{a1})", "name": "beta", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-11,-7,-3,1,5,9)", "name": "c1", "description": ""}, "is_cosec": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(b1*a1<0,true,false)", "name": "is_cosec", "description": ""}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-9..9 except [0,a1])", "name": "b1", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "a1", "description": ""}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "d1", "description": ""}}, "ungrouped_variables": ["a1", "c1", "b1", "d1", "beta", "is_cosec"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Solve a separable first order ODE with trig functions, ", "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>You are given the differential equation</p>\n<p>\\[\\simplify{{a1}*sin(x)*y'}=\\simplify{{b1}*y*cos(x)},\\]</p>\n<p>satisfying $y\\left(\\simplify{{c1}*pi/2}\\right)=\\var{d1}$.</p>\n<p>The solution can be written in the form $y=\\alpha f(x)^\\beta$, where $\\alpha$ and $\\beta$ are constants, with $\\beta&gt;0$, and $f(x)$ is some function of $x$.</p>", "tags": ["checked2015", "MAS1603", "MAS2105"], "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p>Better to ask for solution directly as breaking down the solution in this way forces only one way of inputting.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Find the solution of a first order separable differential equation of the form $a\\sin(x)y'=by\\cos(x)$.</p>"}, "advice": "<p>The differential equation is separable, and we can therefore write</p>\n<p>\\[\\int{\\!\\frac{1}{y}\\,\\mathrm{d}y}=\\simplify{{b1}/{a1}*int(cos(x)/sin(x),x)},\\]</p>\n<p>which can be integrated to give</p>\n<p>\\[\\ln\\lvert y\\rvert=\\simplify{{b1}/{a1}*ln(abs(sin(x)))}+c,\\]</p>\n<p>so</p>\n<p>\\[y=\\simplify[all,fractionnumbers]{A*({if(is_cosec,1,0)}*cosec(x)+{if(is_cosec,0,1)}*sin(x))^({beta})},\\]</p>\n<p>which is the general solution of the equation.</p>\n<p>Then we have</p>\n<p>\\[\\var{d1}=y\\left(\\simplify{{c1}*pi/2}\\right)=\\simplify[all,fractionnumbers]{A*({if(is_cosec,1,0)}*cosec({c1}*pi/2)^({beta})+{if(is_cosec,0,1)}*sin({c1}*pi/2)^({beta}))},\\]</p>\n<p>so $A=\\var{d1}$.</p>\n<p>Then the full solution is</p>\n<p>\\[y=\\simplify[all,fractionnumbers]{{d1}*({if(is_cosec,1,0)}*cosec(x)+{if(is_cosec,0,1)}*sin(x))^({beta})}.\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}