// Numbas version: exam_results_page_options
{"name": "Harry's copy of 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": [{"variable_groups": [], "name": "Harry's copy of Solve a separable first order ODE with trig functions, ", "type": "question", "metadata": {"description": "<p>Find the solution of a first order separable differential equation of the form $a\\sin(x)y'=by\\cos(x)$.</p>", "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"}, "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "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>", "ungrouped_variables": ["a1", "c1", "b1", "d1", "beta", "is_cosec"], "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"is_cosec": {"templateType": "anything", "description": "", "definition": "if(b1*a1<0,true,false)", "group": "Ungrouped variables", "name": "is_cosec"}, "b1": {"templateType": "anything", "description": "", "definition": "random(-9..9 except [0,a1])", "group": "Ungrouped variables", "name": "b1"}, "c1": {"templateType": "anything", "description": "", "definition": "random(-11,-7,-3,1,5,9)", "group": "Ungrouped variables", "name": "c1"}, "beta": {"templateType": "anything", "description": "", "definition": "abs({b1}/{a1})", "group": "Ungrouped variables", "name": "beta"}, "a1": {"templateType": "anything", "description": "", "definition": "random(1..9)*sign(random(-1,1))", "group": "Ungrouped variables", "name": "a1"}, "d1": {"templateType": "anything", "description": "", "definition": "random(1..9)*sign(random(-1,1))", "group": "Ungrouped variables", "name": "d1"}}, "showQuestionGroupNames": false, "functions": {}, "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>", "parts": [{"marks": 0, "variableReplacements": [], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"expectedvariablenames": [], "marks": 1, "answer": "{d1}", "scripts": {}, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "checkingtype": "absdiff", "variableReplacements": [], "type": "jme", "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showpreview": true}, {"expectedvariablenames": [], "marks": 1, "answer": "{beta}", "scripts": {}, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "fractionnumbers", "variableReplacements": [], "type": "jme", "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showpreview": true}, {"expectedvariablenames": [], "marks": 1, "answer": "{if(is_cosec,1,0)}*cosec(x)+{if(is_cosec,0,1)}*sin(x)", "scripts": {}, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "vsetrangepoints": 5, "checkingtype": "absdiff", "variableReplacements": [], "type": "jme", "notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showpreview": 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>"}], "preamble": {"css": "", "js": ""}, "tags": ["checked2015", "MAS1603", "MAS2105"], "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}]}], "contributors": [{"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}]}