// Numbas version: exam_results_page_options {"name": "separable first order ODE with trig functions 3 violeta", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a1", "c1", "b1", "d1", "beta", "is_cosec"], "name": "separable first order ODE with trig functions 3 violeta", "tags": ["checked2015", "MAS1603", "MAS2105"], "type": "question", "preamble": {"css": "", "js": ""}, "question_groups": [{"pickQuestions": 0, "questions": [], "pickingStrategy": "all-ordered", "name": ""}], "showQuestionGroupNames": false, "advice": "

The differential equation is separable, and we can therefore write

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\\[\\int{\\!\\frac{1}{y}\\,\\mathrm{d}y}=\\simplify{{b1}/{a1}*int(cos(x)/sin(x),x)},\\]

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which can be integrated to give

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\\[\\ln\\lvert y\\rvert=\\simplify{{b1}/{a1}*ln(abs(sin(x)))}+c,\\]

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so

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\\[y=\\simplify[all,fractionnumbers]{A*({if(is_cosec,1,0)}*cosec(x)+{if(is_cosec,0,1)}*sin(x))^({beta})},\\]

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which is the general solution of the equation.

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Then we have

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\\[\\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}))},\\]

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so $A=\\var{d1}$.

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Then the full solution is

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\\[y=\\simplify[all,fractionnumbers]{{d1}*({if(is_cosec,1,0)}*cosec(x)+{if(is_cosec,0,1)}*sin(x))^({beta})}.\\]

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Do not enter decimals in your answer.

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Do not enter decimals in your answer.

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Do not enter decimals in your answer.

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Solve the equation, and enter the values of $\\alpha$ and $\\beta$, and the expression for $f(x)$ in the boxes.  Do not enter decimals in your answers.

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$\\alpha=$ [[0]]

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$\\beta=$ [[1]]

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$f(x)=$ [[2]]

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You are given the differential equation

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\\[\\simplify{{a1}*sin(x)*y'}=\\simplify{{b1}*y*cos(x)},\\]

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satisfying $y\\left(\\simplify{{c1}*pi/2}\\right)=\\var{d1}$.

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The solution can be written in the form $y=\\alpha f(x)^\\beta$, where $\\alpha$ and $\\beta$ are constants, with $\\beta>0$, and $f(x)$ is some function of $x$.

", "variables": {"beta": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs({b1}/{a1})", "name": "beta", "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": ""}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "d1", "description": ""}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(-11,-7,-3,1,5,9)", "name": "c1", "description": ""}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "name": "a1", "description": ""}}, "metadata": {"notes": "

Better to ask for solution directly as breaking down the solution in this way forces only one way of inputting.

", "description": "

Find the solution of a first order separable differential equation of the form $a\\sin(x)y'=by\\cos(x)$.

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