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$$
\begin{array}{l}{\text { No energy is stored in the } 0.1 \mathrm{H} \text { inductor or the }} \\ {0.4 \mu \mathrm{F} \text { capacitor when the switch in the circuit }} \\ {\text { shown is closed. Find } v_{c}(t) \text { for } t \geq 0 \text { . }}\end{array}
$$

$$
S^{2}+\frac{R}{L} S+\frac{1}{L C}=0
$$

$$
S^{2}+\frac{1000 s}{0.1}+\frac{1}{0.1*0 .4*10^{-6}}=0
$$

$$
S^{2}+10000 $+25 \times 10^{6}=0
$$

$$
S_{1}=-5000 \quad , S_{2}=-5000
$$

$$
-V e
$$
$$
\text { repeated, real } \Rightarrow \text {critically damped}
$$

$$
V_c(t)=V_{final}+(A_1 t+A_2)e^{s_1 t}
$$

DC

$$
\frac{d v}{d t}=0 \Rightarrow C^{t}_{c}=0 \Rightarrow0.C
$$

KVL

$$
V_{f}=48 volt
$$

$$
w=\frac{1}{2} c v^{2} \Rightarrow V=0 \quad V(0)=0
$$

$$
V(0)=0=V_f+A_{2}
$$

$$
A_{2}=-V_f=-48v
$$

$$
\frac{d v}{d t}=0=-5000 A_2+A_1
$$

$$
A_1=-24000
$$

$$
V_{c} (t)=48+[-24000 t+-48) e^{-5000  t} \quad (V)
$$

$$
t \geq 0
$$

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