12. (a) A series circuit has a capacitor of 1E−5 farad, a resistor of 3E2 ohms, and an inductor of
0.2 henry. The initial charge on the capacitor
is 1E−6 coulomb and there is no initial current.
Find the charge q on the capacitor at any time t.
(b) Draw a phase portrait of the equivalent dynamical system that includes the trajectory
corresponding to the initial value problem in
part (a).

Question

12. (a) A series circuit has a capacitor of 1E−5 farad, a resistor of 3E2 ohms, and an inductor of
0.2 henry. The initial charge on the capacitor
is 1E−6 coulomb and there is no initial current.
Find the charge q on the capacitor at any time t.
(b) Draw a phase portrait of the equivalent dynamical system that includes the trajectory
corresponding to the initial value problem in
part (a).

anonymous · Answer

$$Ldi/dt + Ri + (1/C)q = e(t)$$

anonymous · Answer

$$ i_{0}^{'}= \frac{e(t_0) - Ri_0 - (1/C)q_0 }{ L }$$

anonymous · Answer

$$\frac{ d ^{2} Q}{ dt ^{2} } + \frac{ R }{ L }\frac{ dQ}{ dt  } + \frac{ 1 }{LC }Q=0$$
$$\lambda ^{2} +\frac{ R }{ L }\lambda +\frac{ 1 }{ LC } = 0$$

anonymous · Answer

Forgot to say thanks, that helped.