Question

An electrical circuit contains an electromotive force \(E\) (supplied by a battery or generator), a resistor \(R\), an inductor \(L\), and a capacitor \(C\), in series. If the charge on the capacitor at time \(t\) is \(Q-Q(t)\), then the current is the rate of change of \(Q\) with respect to \(t: I=\frac{H Q}{d w}\). Kirchhoff s voltage law gives the supplied voltage as:

$$ L \frac{d i}{w}+R t+\frac{R}{E}=E(t) $$

Since \(I=\frac{1}{1}\), the differential equation becomes:

$$ L Q^{\prime \prime}+R Q^{\prime}+\frac{1}{c} Q=E(0) $$

with initial conditions \(Q(0)=Q_{0}\) and \(Q(0)=I(0)=I_{0}\)

(a) If \(R=30\) ohms, \(L=1\) henry, \(C=11.764705 \times 10^{-4}\) far ads, and \(E(t)=70 \cdot \cos (7 t)\), what is the general solution to the complementary equation \(q_{C}(t) ?\) (10 points)

(b) Use the method of undetermined coefficients to find a particular solution, \(Q_{p}(t)\), for the non-homogeneous equation. For undetermined coefficients \(A\) \& \(B\) in your guess, reduce the decimal to 3 places and convert to a fraction (10 points)

(c) If \(c_{1}=\frac{-14}{697}\) and \(c_{2}=\frac{44}{2(x)}\) in the complementary solution, find the charge \(Q(t)\), and current \(I(t)\). Reeall that \(Q(t)=q_{C}(t)+Q_{p}(t)\) and \(I(t)=Q(t)\). (You do not need all the space provided to answer this: question, just write out the full solutions) (10 points)

Answer 1