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{"name": "Differential equation with an irreducible quadratic factor 2: X(s)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["R", "b1", "b2", "a1", "a2"], "variablesTest": {"maxRuns": "197", "condition": ""}, "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>", "variable_groups": [], "statement": "<p>The solution to the differential equation:</p>\n<p>&nbsp; &nbsp; &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>is given by</p>\n<p>&nbsp; &nbsp; &nbsp;\\(x(t)=A+Be^{-\\var{a1}t}cos(\\var{a2}t)+Ce^{-\\var{a1}t}sin(\\var{a2}t)\\)</p>", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "<p>Solve a&nbsp;Differential equation with an irreducible quadratic factor&nbsp;</p>"}, "functions": {}, "preamble": {"js": "", "css": ""}, "tags": [], "variables": {"R": {"name": "R", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..20)"}, "b1": {"name": "b1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..12)"}, "a2": {"name": "a2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(2..12)"}, "b2": {"name": "b2", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..12)"}, "a1": {"name": "a1", "description": "", "templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..10)"}}, "parts": [{"precisionType": "dp", "maxValue": "{R}/({a1}*{a1}+{a2}*{a2})", "showCorrectAnswer": true, "strictPrecision": true, "correctAnswerFraction": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "showPrecisionHint": false, "type": "numberentry", "mustBeReduced": false, "prompt": "<p>Enter the value for \\(A\\) correct to three decimal places.</p>", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "precision": "3", "minValue": "{R}/({a1}*{a1}+{a2}*{a2})", "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionType": "dp", "maxValue": "{b1}-{R}/({a1}*{a1}+{a2}*{a2})", "showCorrectAnswer": true, "strictPrecision": true, "correctAnswerFraction": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "showPrecisionHint": false, "type": "numberentry", "mustBeReduced": false, "prompt": "<p>Enter the value for \\(B\\) correct to three decimal places.</p>", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "precision": "3", "minValue": "{b1}-{R}/({a1}*{a1}+{a2}*{a2})", "precisionMessage": "You have not given your answer to the correct precision."}, {"precisionType": "dp", "maxValue": "(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a2}*({a1}^2+{a2}^2))", "showCorrectAnswer": true, "strictPrecision": true, "correctAnswerFraction": false, "precisionPartialCredit": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacements": [], "marks": 1, "showPrecisionHint": false, "type": "numberentry", "mustBeReduced": false, "prompt": "<p>Enter the value for \\(C\\) correct to three decimal places.</p>", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "precision": "3", "minValue": "(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R}-{a1}*({b1}*({a1}^2+{a2}^2)-{R}))/({a2}*({a1}^2+{a2}^2))", "precisionMessage": "You have not given your answer to the correct precision."}], "name": "Differential equation with an irreducible quadratic factor 2: X(s)", "extensions": [], "rulesets": {}, "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/"}]}