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{"name": "Application 2: Differential equation with an irreducible quadratic factor: X(s)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Solve a&nbsp;Differential equation with an irreducible quadratic factor&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "variable_groups": [], "statement": "<p>A metal plate attached to a stiff spring and immersed in viscous oil is in motion and its position \\(x(t)\\) at time \\(t\\) satisfies the differential equation</p>\n<p style=\"padding-left: 30px;\">&nbsp; &nbsp; &nbsp;\\(\\frac{d^2x}{dt^2}+\\simplify{2*{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}x(t)=\\var{R}\\) &nbsp;</p>\n<p>The plate is in moving at \\(\\var{b2}cm/s\\) and is initially at a height of \\(\\var{b1}cm\\) so that, \\(x(0)=\\var{b1} \\,\\, and \\,\\, &nbsp;x'(0)=\\var{b2}\\)</p>\n<p>The solution to the differential equation is given by</p>\n<p style=\"padding-left: 30px;\">&nbsp; &nbsp;&nbsp;&nbsp;\\(x(t)=A+Be^{-\\var{a1}t}cos(\\var{a2}t)+Ce^{-\\var{a1}t}sin(\\var{a2}t)\\)</p>\n<p>.</p>", "preamble": {"js": "", "css": ""}, "ungrouped_variables": ["R", "b1", "b2", "a1", "a2", "a", "b", "c"], "name": "Application 2: Differential equation with an irreducible quadratic factor: X(s)", "tags": [], "variables": {"b2": {"definition": "random(1..12)", "group": "Ungrouped variables", "templateType": "anything", "name": "b2", "description": ""}, "R": {"definition": "random(1..20)", "group": "Ungrouped variables", "templateType": "anything", "name": "R", "description": ""}, "a2": {"definition": "random(2..12)", "group": "Ungrouped variables", "templateType": "anything", "name": "a2", "description": ""}, "b": {"definition": "{b1}-{R}/({a1}*{a1}+{a2}*{a2})", "group": "Ungrouped variables", "templateType": "anything", "name": "b", "description": ""}, "a": {"definition": "{R}/({a1}*{a1}+{a2}*{a2})", "group": "Ungrouped variables", "templateType": "anything", "name": "a", "description": ""}, "a1": {"definition": "random(1..10)", "group": "Ungrouped variables", "templateType": "anything", "name": "a1", "description": ""}, "b1": {"definition": "random(1..12)", "group": "Ungrouped variables", "templateType": "anything", "name": "b1", "description": ""}, "c": {"definition": "(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a2}*({a1}^2+{a2}^2))", "group": "Ungrouped variables", "templateType": "anything", "name": "c", "description": ""}}, "parts": [{"scripts": {}, "type": "gapfill", "prompt": "<p>Enter the value for \\(A\\) as an exact fraction.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(A=\\)&nbsp;[[0]]</p>\n<p>Enter the value for \\(B\\) as an exact fraction.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(B=\\) &nbsp;[[1]]</p>\n<p>Enter the value for \\(C\\) as an exact fraction.&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(C=\\) &nbsp;[[2]]</p>", "variableReplacements": [], "gaps": [{"type": "numberentry", "minValue": "{R}/({a1}*{a1}+{a2}*{a2})", "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "maxValue": "{R}/({a1}*{a1}+{a2}*{a2})", "scripts": {}, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "marks": 1, "correctAnswerStyle": "plain"}, {"type": "numberentry", "minValue": "{b1}-{R}/({a1}*{a1}+{a2}*{a2})", "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "maxValue": "{b1}-{R}/({a1}*{a1}+{a2}*{a2})", "scripts": {}, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "marks": 1, "correctAnswerStyle": "plain"}, {"type": "numberentry", "minValue": "(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a2}*({a1}^2+{a2}^2))", "allowFractions": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "mustBeReducedPC": 0, "maxValue": "(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a2}*({a1}^2+{a2}^2))", "scripts": {}, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "mustBeReduced": false, "correctAnswerFraction": true, "marks": 1, "correctAnswerStyle": "plain"}], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true}], "extensions": [], "advice": "<p>&nbsp;\\(\\frac{d^2x}{dt^2}+\\simplify{2*{a1}}\\frac{dx}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}x(t)=\\var{R}\\) &nbsp;where \\(x(0)=\\var{b1} \\,\\, and \\,\\, &nbsp;x'(0)=\\var{b2}\\)</p>\n<p>The Laplace transform of this is given by:</p>\n<p>\\(s^2X(s)-\\var{b1}s-\\var{b2}+\\simplify{2*{a1}}(sX(s)-\\var{b1})+\\simplify{{a1}*{a1}+{a2}*{a2}}X(s)=\\frac{\\var{R}}{s}\\)</p>\n<p>Gathering only \\(X(s)\\) terms on the left hand side and factoring gives:</p>\n<p>\\(s^2X(s)+\\simplify{2*{a1}}sX(s)+\\simplify{{a1}^2+{a2}^2}X(s)=\\frac{\\var{R}}{s}+\\var{b1}s+\\simplify{{b2}+2*{a1}*{b1}}\\)</p>\n<p>\\((s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})X(s)=\\frac{\\var{b1}s^2+\\simplify{{b2}+2*{a1}*{b1}}s+\\var{R}}{s}\\)</p>\n<p>\\(X(s)=\\frac{\\var{b1}s^2+\\simplify{{b2}+2*{a1}*{b1}}s+\\var{R}}{s(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\\)</p>\n<p>\\(X(s)=\\frac{A}{s}+\\frac{Bs+c}{(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\\)</p>\n<p>\\(\\var{b1}s^2+\\simplify{{b2}+2*{a1}*{b1}}s+\\var{R}=A(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})+Bs^2+cs\\)</p>\n<p>Let \\(s=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{R}=A(\\simplify{{a1}^2+{a2}^2})\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\simplify{{R}/({a1}^2+{a2}^2)}\\)</p>\n<p>Compare the coefficients of \\(s^2\\) &nbsp;&nbsp;</p>\n<p style=\"padding-left: 30px;\">\\(\\var{b1}=A+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\var{b1}-\\simplify{{R}/({a1}^2+{a2}^2)}=\\simplify{({b1}*({a1}^2+{a2}^2)-{R})/({a1}^2+{a2}^2)}\\)</p>\n<p>Compare the coefficients of \\(s\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{b2}+2*{a1}*{b1}}=\\simplify{2*{a1}}A+c\\)</p>\n<p style=\"padding-left: 30px;\">\\(c=\\simplify{{b2}+2*{a1}*{b1}}-\\simplify{2*{a1}}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(c=\\simplify{(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R})/({a1}^2+{a2}^2)}\\)</p>\n<p style=\"padding-left: 30px;\"></p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{\\simplify{{R}/({a1}^2+{a2}^2)}}{s}+\\frac{\\simplify{({b1}*({a1}^2+{a2}^2)-{R})/({a1}^2+{a2}^2)}s+\\simplify{(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R})/({a1}^2+{a2}^2)}}{s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2}}\\)</p>\n<p style=\"padding-left: 30px;\">\\(X(s)=\\frac{\\simplify{{R}/({a1}^2+{a2}^2)}}{s}+\\frac{\\frac{\\simplify{({b1}*({a1}^2+{a2}^2)-{R})}}{\\simplify{({a1}^2+{a2}^2)}}(s+\\var{a1})}{(s+\\var{a1})^2+\\var{a2}^2}+\\frac{\\simplify{(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a1}^2+{a2}^2)}}{(s+\\var{a1})^2+\\var{a2}^2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x(t)=\\simplify{{R}/({a1}^2+{a2}^2)}+\\simplify{({b1}*({a1}^2+{a2}^2)-{R})/({a1}^2+{a2}^2)}e^{-\\var{a1}t}cos(\\var{a2}t)+\\simplify{(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a2}*({a1}^2+{a2}^2))}e^{-\\var{a1}t}sin(\\var{a2}t)\\)</p>\n<p></p>", "rulesets": {}, "variablesTest": {"maxRuns": "197", "condition": ""}, "functions": {}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}