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The solution of the equation is given by

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\$$q(t)=A+Be^{-\\var{a1}t}cos(\\var{a2}t)+Ce^{-\\var{a1}t}sin(\\var{a2}t)\$$

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Enter the value for \$$A\$$ correct to three decimal places.   \$$A=\$$ [[0]]

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Enter the value for \$$B\$$ correct to three decimal places.   \$$B=\$$ [[1]]

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Enter the value for \$$C\$$ correct to three decimal places.   \$$C=\$$ [[2]]

\$$\\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}\$$

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The Laplace transform of this is given by:

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\$$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}\$$

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Gathering only \$$I(s)\$$ terms on the left hand side and factoring gives:

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\$$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}}\$$

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\$$(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}\$$

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\$$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})}\$$

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\$$Q(s)=\\frac{A}{s}+\\frac{Bs+c}{(s^2+\\simplify{2*{a1}}s+\\simplify{{a1}^2+{a2}^2})}\$$

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\$$\\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\$$

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Let \$$s=0\$$

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\$$\\var{R}=A(\\simplify{{a1}^2+{a2}^2})\$$

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\$$A=\\simplify{{R}/({a1}^2+{a2}^2)}\$$

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Compare the coefficients of \$$s^2\$$

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\$$\\var{b1}=A+B\$$

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\$$B=\\var{b1}-\\simplify{{R}/({a1}^2+{a2}^2)}=\\simplify{({b1}*({a1}^2+{a2}^2)-{R})/({a1}^2+{a2}^2)}\$$

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Compare the coefficients of \$$s\$$

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\$$\\simplify{{b2}+2*{a1}*{b1}}=\\simplify{2*{a1}}A+c\$$

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\$$c=\\simplify{{b2}+2*{a1}*{b1}}-\\simplify{2*{a1}}A\$$

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\$$c=\\simplify{(({b2}+2*{a1}*{b1})*({a1}^2+{a2}^2)-2*{a1}*{R})/({a1}^2+{a2}^2)}\$$

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\$$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}}\$$

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\$$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}\$$

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\$$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)\$$

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", "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)\$$.

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Prior to closing the switch at time \$$t\$$ = 0, both the charge on the capacitor and the resulting current in the circuit are zero.

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The following equation may be derived using Kirchoff's Voltage law.

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\$$L\\frac{di}{dt}+Ri(t)+\\frac{1}{C}q(t)=e(t)\$$

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If \$$L=1\$$,   \$$R=\\simplify{2*{a1}}\$$,    \$$C=\\frac{1}{\\simplify{{a1}*{a1}+{a2}*{a2}}}\$$  and  \$$e(t)=\\var{R}\$$

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this will give rise to the differential equation:

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\$$\\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}\$$

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", "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/"}]}