A capacitor of capacitance charges through a resistor from a battery of

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Problem

A capacitor of capacitance C=2.0×10−6 FC=2.0\times10^{-6}\,\mathrm F charges through a resistor R=1000 ΩR=1000\,\Omega from a battery of V0=9 VV_0=9\,\mathrm V, obeying RdQdt+QC=V0,Q(0)=0.R\frac{\mathrm dQ}{\mathrm dt}+\frac{Q}{C}=V_0,\qquad Q(0)=0. A symbolic solver returns Q(t)=CV0(1−e−t/(RC)).Q(t)=CV_0\left(1-e^{-t/(RC)}\right). ((a)) Verify by direct substitution that this Q(t)Q(t) satisfies the ODE for all t≥0t\ge0 and satisfies the initial condition, stating the residual obtained. ((b)) State the final charge Q(t→∞)Q(t\to\infty) in coulombs.

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