Determine the current as a function of time $t=0$ for the circuit shown below.

CASE $1$:

When

$(\frac{R}{L}{)}^2>\frac{4}{LC}$

$m_1=\frac{\frac{-R}{L}+\sqrt[\phantom{2}]{(\frac{R}{L}{)}^2-\frac{4}{LC}}}{2}$

$m_2=\frac{\frac{-R}{L}-\sqrt[\phantom{2}]{(\frac{R}{L}{)}^2-\frac{4}{LC}}}{2}$

$i=C_1{e}^{m1t}+C_2{e}^{m2t}$

CASE $2$:

When

$(\frac{R}{L}{)}^2=\frac{4}{LC}$

$m_1=m_2=\frac{-R}{2L}$

$i=C_1{e}^{mlt}+C_{2}t{e}^{mlt}$

$A=\frac{-R}{2L}$

$B=\frac{\sqrt[\phantom{2}]{(\frac{4}{LC})-(\frac{R}{L}{)}^2}}{2}$

$i=C_1{e}^{mlt}+C_2{e}^{m2t}$

$i=C_1{e}^{(A+jB)t}+C_2{e}^{(A-jB)t}$Assignment # $1$

Determine the current as a function of time $t=0$ for the circuit shown below.