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\\(\\frac{d^2q}{dt^2}+\\simplify{{a1}+{b1}}\\frac{dq}{dt}+\\simplify{{a1}*{b1}}q(t)=\\var{c1}e^{-\\var{d1}t}\\)  where \\(q(0)=\\var{i0} \\,\\, and \\,\\,  q'(0)=\\var{i1}\\)

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The Laplace transform of this is given by:

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\\(s^2Q(s)-\\var{i0}s-\\var{i1}+\\simplify{{a1}+{b1}}(sQ(s)-\\var{i0})+\\simplify{{a1}*{b1}}Q(s)=\\frac{\\var{c1}}{s+\\var{d1}}\\)

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

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\\((s^2+\\simplify{{a1}+{b1}}s+\\simplify{{a1}*{b1}})Q(s)=\\frac{\\var{c1}}{s+\\var{d1}}+\\var{i0}s+\\simplify{{i1}+({a1}+{b1})*{i0}}\\)

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\\((s^2+\\simplify{{a1}+{b1}}s+\\simplify{{a1}*{b1}})Q(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{b1})*{i0})*(s+{d1})}}{s+\\var{d1}}\\)

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\\(Q(s)=\\frac{\\simplify{{c1}+({i0}s+{i1}+({a1}+{b1})*{i0})*(s+{d1})}}{(s+\\var{d1})(s+\\var{a1})(s+\\var{b1})}\\)

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\\(Q(s)=\\frac{A}{s+\\var{d1}}+\\frac{B}{s+\\var{a1}}+\\frac{C}{s+\\var{b1}}\\)

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\\(\\simplify{{c1}+({i0}s+{i1}+({a1}+{b1})*{i0})*(s+{d1})}=A(s+\\var{a1})(s+\\var{b1})+B(s+\\var{d1})(s+\\var{b1})+C(s+\\var{d1})(s+\\var{a1})\\)

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Let \\(s=-\\var{d1}\\)

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\\(\\simplify{{c1}+({i0}*-{d1}+{i1}+({a1}+{b1})*{i0})*(-{d1}+{d1})}=\\simplify{(-{d1}+{a1})(-{d1}+{b1})}A\\)

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\\(A=\\simplify{({c1}+({i0}*-{d1}+{i1}+({a1}+{b1})*{i0})*(-{d1}+{d1}))/((-{d1}+{a1})(-{d1}+{b1}))}\\)

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Let \\(s=-\\var{a1}\\)

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\\(\\simplify{{c1}+({i0}*-{a1}+{i1}+({a1}+{b1})*{i0})*(-{a1}+{d1})}=\\simplify{(-{a1}+{d1})(-{a1}+{b1})}B\\)

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\\(B=\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{b1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1})(-{a1}+{b1}))}\\)

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Let \\(s=-\\var{b1}\\)

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\\(\\simplify{{c1}+({i0}*-{b1}+{i1}+({a1}+{b1})*{i0})*(-{b1}+{d1})}=\\simplify{(-{b1}+{d1})(-{b1}+{a1})}C\\)

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\\(C=\\simplify{({c1}+({i0}*-{b1}+{i1}+({a1}+{b1})*{i0})*(-{b1}+{d1}))/((-{b1}+{d1})(-{b1}+{a1}))}\\)

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\\(q(t)=\\simplify{({c1}+({i0}*-{d1}+{i1}+({a1}+{b1})*{i0})*(-{d1}+{d1}))/((-{d1}+{a1})(-{d1}+{b1}))}e^{-\\var{d1}t}+\\simplify{({c1}+({i0}*-{a1}+{i1}+({a1}+{b1})*{i0})*(-{a1}+{d1}))/((-{a1}+{d1})(-{a1}+{b1}))}e^{-\\var{a1}t}+\\simplify{({c1}+({i0}*-{b1}+{i1}+({a1}+{b1})*{i0})*(-{b1}+{d1}))/((-{b1}+{d1})(-{b1}+{a1}))}e^{-\\var{b1}t}\\)

", "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{d^2q}{dt^2}+R\\frac{dq}{dt}+\\frac{1}{C}q(t)=e(t)\\)

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Given \\(L=1Henry,\\,\\,R=\\simplify{{a1}+{b1}}ohms\\,\\,and\\,\\,C=\\frac{1}{\\simplify{{a1}*{b1}}}Farad\\)

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Laplace transforms can be used to determine the charge \\(q(t)\\) on the capacitor at time t.

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 The solution to the differential equation:

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     \\(\\frac{d^2q}{dt^2}+\\simplify{{a1}+{b1}}\\frac{dq}{dt}+\\simplify{{a1}*{b1}}q(t)=\\var{c1}e^{-\\var{d1}t}\\)  where \\(q(0)=\\var{i0} \\,\\, and \\,\\,  q'(0)=\\var{i1}\\)

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is given by

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     \\(q(t)=Ae^{-\\var{d1}t}+Be^{-\\var{a1}t}+Ce^{-\\var{b1}t}\\)

", "name": "Application 3:Differential equation with 3 simple linear factors: Q(s)", "metadata": {"description": "

Solve a Differential equation with 3 simple linear factors

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Enter the value for \\(A\\) correct to three decimal places.

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Enter the value for \\(B\\) correct to three decimal places.

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Enter the value for \\(C\\) correct to three decimal places.

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