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Kirchoff's Voltage Law:
$V_0 = V_L + V_R + V = L\frac{dI_1}{dt} + IR + V$
Current conservation at node X:
$I_1 = I_2 + I_3 $
$I_2 = I_C = C\frac{dV}{dt}$
$I_3 = \frac{V}{R}$
$I_1 = C\frac{dV}{dt} + \frac{V}{R} $
$\frac{dI_1}{dt} = C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}$
$V_0 = L(C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}) + (C\frac{dV}{dt} + \frac{V}{R})R + V = LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V$
The governing differential equation is:
$LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V = V_0 $
Differential Electricity
Age 16 to 18
Challenge Level 
Kirchoff's Voltage Law:
$V_0 = V_L + V_R + V = L\frac{dI_1}{dt} + IR + V$
Current conservation at node X:
$I_1 = I_2 + I_3 $
$I_2 = I_C = C\frac{dV}{dt}$
$I_3 = \frac{V}{R}$
$I_1 = C\frac{dV}{dt} + \frac{V}{R} $
$\frac{dI_1}{dt} = C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}$
$V_0 = L(C\frac{d^2V}{dt^2} +\frac{1}{R} \frac{dV}{dt}) + (C\frac{dV}{dt} + \frac{V}{R})R + V = LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V$
The governing differential equation is:
$LC \frac{d^2V}{dt^2} + (\frac{L}{R} + CR)\frac{dV}{dt} + 2V = V_0 $
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