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$$
\begin{array}{l}{\text { For the circuit shown, } v\left(0^{+}\right)=12 \mathrm{V}, \text { and } i_{l}\left(0^{+}\right)=30 \mathrm{mA}} \\ {\text { a) Find the initial current in each branch of the circuit. }} \\ {\text { b) Find the initial value of } d v / d t} \\ {\text { c) Find the expression for } v(t)}\end{array}
$$

$$
i_{l}(0)=30 \mathrm{mA}
$$

$$
i_{R}(0)=\frac{V_{0}(0)}{R}=\frac{12}{200}=60 \mathrm{mA}
$$

KCL

$$
I_{c}+I_{R}+I_{c}=0
$$

$$
I_{c}(0)=I_{R}(0)-I_{c}(0)
$$

$$
I_{C}(0)=-30-60=-90 \mathrm{mA}
$$

$$
i_{c}= C \frac{d v(0)}{d t}
$$

$$
-90 \times 10^{-3}=0.2 \times 10^{-6}* \frac{dv(0)}{d t}
$$

$$
\frac{d v(0)}{d t}=
-l150 \ kv/sec
$$

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

$$
S^{2}+\frac{S}{200 \times 0.2 \times 10^{-6}}+\frac{1}{50 \times 10^{-3} \times 0.2 \times 10^{-6}}=0
$$

$$
S^{2}+25000S+10^{8}=0
$$

$$
S_{1}=-5000 \quad , S_{2}=-20000
$$

$$
\text {*real*} -V e * S_{1} \neq S_{2} \Rightarrow \text { overdomped }
$$

$$
V(t)=A_{1} e^{s_1 t}+A_2e^{s_2 t}
$$

$$
V(t)=A_1 e^{-5000 t}+A_{2} e^{-20000t}
$$

$$
V(0)=12 \mathrm{V} \quad \frac{d V(0)}{d t}=-450*10^3 \mathrm{V} / \mathrm{sec}
$$

$$
t=0 \quad V(0)=12 V \Rightarrow V(t)=A_1 e^{-5000 t}+A_{2} e^{-20000 t}
$$

$$
12=A_{1}+A_{2} \rightarrow (1)
$$

$$
\frac{d v(t)}{d t}=-5000 A_1 \mathrm{e}^{-5000t}+-20000 \mathrm{A_2 e}^{-20000t}
$$

$$
\frac{d v(0)}{d t}=-450 \times 10^{3} \mathrm{v} / \mathrm{sec}
$$                        $$
t=0 \Rightarrow
$$

$$
-450 \times 10^{-3}=-5000 A_{1}-20000 \mathrm{A_2}\rightarrow(2)
$$

$$
A_1=-14 V
$$                $$
A_{2}=26 \mathrm{V}
$$

$$
V(t)=-14e^{-5000 t}+26 e^{-20000 t}
$$

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