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The charge \(q(t)\) (in coulombs) on the capacitor in a series RLC circuit satisfies the differential equation$$ 20 \dfrac{d^2 q}{dt^2} + 10 \dfrac{dq}{dt} + 100 q = 0, $$where \(t\) is time in seconds.
  1. Use the substitution \(i = \dfrac{dq}{dt}\) (where \(i\) represents the current in the circuit) to rewrite this equation as a coupled system of first-order differential equations.
  2. The equation for \(\dfrac{di}{dt}\) is separable and independent of \(q\). Solve this equation to find \(i(t)\).
  3. Hence obtain the general solution for \(q(t)\).

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