// Numbas version: exam_results_page_options {"name": "Differential equation with an irreducible quadratic factor 2: Q(s)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/RLC.jpg", "/srv/numbas/media/question-resources/RLC.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "scripts": {}, "prompt": "
The solution of the equation is given by
\n\\(q(t)=A+Be^{-\\var{a1}t}cos(\\var{a2}t)+Ce^{-\\var{a1}t}sin(\\var{a2}t)\\)
\nEnter the value for \\(A\\) correct to three decimal places. \\(A=\\) [[0]]
\nEnter the value for \\(B\\) correct to three decimal places. \\(B=\\) [[1]]
\nEnter the value for \\(C\\) correct to three decimal places. \\(C=\\) [[2]]
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\nThe Laplace transform of this is given by:
\n\\(s^2Q(s)-\\var{b1}s-\\var{b2}+\\simplify{2*{a1}}(sQ(s)-\\var{b1})+\\simplify{{a1}*{a1}+{a2}*{a2}}Q(s)=\\frac{\\var{R}}{s}\\)
\nGathering only \\(I(s)\\) terms on the left hand side and factoring gives:
\n\\(s^2I(s)+\\simplify{2*{a1}}sI(s)+\\simplify{{a1}^2+{a2}^2})Q(s)=\\frac{\\var{R}}{s}+\\var{b1}s+\\simplify{{b2}+2*{a1}*{b1}}\\)
\n\\((s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})Q(s)=\\frac{\\var{b1}s^2+\\simplify{{b2}+2*{a1}*{b1}}s+\\var{R}}{s}\\)
\n\\(Q(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})}\\)
\n\\(Q(s)=\\frac{A}{s}+\\frac{Bs+c}{(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\\)
\n\\(\\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\\)
\nLet \\(s=0\\)
\n\\(\\var{R}=A(\\simplify{{a1}^2+{a2}^2})\\)
\n\\(A=\\simplify{{R}/({a1}^2+{a2}^2)}\\)
\nCompare the coefficients of \\(s^2\\)
\n\\(\\var{b1}=A+B\\)
\n\\(B=\\var{b1}-\\simplify{{R}/({a1}^2+{a2}^2)}=\\simplify{({b1}*({a1}^2+{a2}^2)-{R})/({a1}^2+{a2}^2)}\\)
\nCompare the coefficients of \\(s\\)
\n\\(\\simplify{{b2}+2*{a1}*{b1}}=\\simplify{2*{a1}}A+c\\)
\n\\(c=\\simplify{{b2}+2*{a1}*{b1}}-\\simplify{2*{a1}}A\\)
\n\\(c=\\simplify{(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R})/({a1}^2+{a2}^2)}\\)
\n\n\\(Q(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}}\\)
\n\\(Q(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}\\)
\n\\(q(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)\\)
\n", "name": "Differential equation with an irreducible quadratic factor 2: Q(s)", "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": "197"}, "extensions": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": "Solve a Differential equation with an irreducible quadratic factor
"}, "rulesets": {}, "statement": "An RLC circuit consists of a resistor R, a capacitor C and an inductor L connected in series with a voltage source \\(e(t)\\).
\nPrior to closing the switch at time \\(t\\) = 0, both the charge on the capacitor and the resulting current in the circuit are zero.
\n\nThe following equation may be derived using Kirchoff's Voltage law.
\n\\(L\\frac{di}{dt}+Ri(t)+\\frac{1}{C}q(t)=e(t)\\)
\nIf \\(L=1\\), \\(R=\\simplify{2*{a1}}\\), \\(C=\\frac{1}{\\simplify{{a1}*{a1}+{a2}*{a2}}}\\) and \\(e(t)=\\var{R}\\)
\nthis will give rise to the differential equation:
\n\\(\\frac{d^2q}{dt^2}+\\simplify{2*{a1}}\\frac{dq}{dt}+\\simplify{{a1}*{a1}+{a2}*{a2}}q(t)=\\var{R}\\) where \\(q(0)=\\var{b1} \\,\\, and \\,\\, q'(0)=\\var{b2}\\)
\n", "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/"}]}