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{"name": "Separable variables (1) ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a1", "c1", "b1", "d1"], "name": "Separable variables (1) ", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>The differential equation is separable, so we can write</p>\n<p>\\[\\int{\\!\\frac{1}{\\var{b1}+y}\\,\\mathrm{d}y} = \\int{\\!\\frac{1}{\\var{a1}+x}\\,\\mathrm{d}x},\\]</p>\n<p>then</p>\n<p>\\[\\ln\\lvert\\var{b1}+y\\rvert=\\ln\\lvert\\var{a1}+x\\rvert+c,\\]</p>\n<p>so</p>\n<p>\\[y=\\simplify{A({a1}+x)-{b1}},\\]</p>\n<p>which is the general solution of the equation.</p>\n<p>Now,</p>\n<p>\\[\\var{d1}=y(\\var{c1})=\\simplify[std]{A({a1}+{c1})-{b1}},\\]</p>\n<p>so</p>\n<p>\\[A=\\simplify[std]{({d1}+{b1})/({a1}+{c1})}=\\simplify{{d1+b1}/{a1+c1}},\\]</p>\n<p>and then the full solution is</p>\n<p>\\[y=\\simplify[std]{{d1+b1}/{a1+c1}({a1}+x)-{b1}}=\\simplify{{(d1*a1-b1*c1)}/{a1+c1}+{(d1+b1)}*x/{a1+c1}}.\\]</p>", "rulesets": {"std": ["all", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "<p>$y=$ [[0]] (Do not enter decimals in your answer.)</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notallowed": {"message": "<p>Do not enter decimals in your answer.</p>", "showStrings": false, "strings": ["."], "partialCredit": 0}, "vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{(d1*a1-b1*c1)}/{a1+c1}+{(d1+b1)}*x/{a1+c1}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Equations which can be written in the form</p>\n<p>\\[\\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = f(x),&nbsp;&nbsp; \\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = f(y),&nbsp;&nbsp; \\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = f(x)f(y)\\]</p>\n<p>can all be solved by integration.</p>\n<p>In each case it is possible to separate the $x$'s to one side of the equation and the $y$'s to the other</p>\n<p>Solving such equations is therefore known as solution by <strong>separation of variables</strong></p>\n<p></p>\n<p><span style=\"text-decoration: underline;\"><strong>Question</strong></span></p>\n<p>Find the solution of the differential equation</p>\n<p>\\[(\\var{a1}+x)\\dfrac{\\mathrm{d}y}{\\mathrm{d}x}=\\var{b1}+y\\]</p>\n<p></p>\n<p>satisfying the condition that $y = \\var{d1}$ when $x = \\var{c1}$</p>\n<p></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a1": {"definition": "random(1..9)*sign(random(-1,1))", "templateType": "anything", "group": "Ungrouped variables", "name": "a1", "description": ""}, "c1": {"definition": "random(1..9 except abs(a1))*sign(random(-1,1))", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "b1": {"definition": "random(1..9)*sign(random(-1,1))", "templateType": "anything", "group": "Ungrouped variables", "name": "b1", "description": ""}, "d1": {"definition": "random(1..9 except abs(b1))*sign(random(-1,1))", "templateType": "anything", "group": "Ungrouped variables", "name": "d1", "description": ""}}, "metadata": {"description": "<p>Equations which can be written in the form</p>\n<p>\\[\\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = f(x),&nbsp;&nbsp; \\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = f(y),&nbsp;&nbsp; \\dfrac{\\mathrm{d}y}{\\mathrm{d}x} = f(x)f(y)\\]</p>\n<p>can all be solved by integration.</p>\n<p>In each case it is possible to separate the $x$'s to one side of the equation and the $y$'s to the other</p>\n<p>Solving such equations is therefore known as solution by <strong>separation of variables</strong></p>\n<p><strong></strong></p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Stephen Bowlzer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/206/"}]}]}], "contributors": [{"name": "Stephen Bowlzer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/206/"}]}