The input/output differential equation for the circuit is Vc'' + R1/L * Vc' + Vc/LC = Vin/L
we find the input/output differential equation for the circuit by using Kirchhoff's voltage law (KVL) to write an equation for the voltage across the capacitor (Vc):
Vc = Vin - R1 * Il - L * dIl/dt
Vin= the input voltage
R1 = the resistance of the resistor
L = the inductance of the inductor
Il = the current flowing through the inductor
We also write an equation for the current flowing through the capacitor:
Ic = C * dVc/dt
C = the capacitance of the capacitor
The inductor and capacitor are connected in parallel, the current flowing through the inductor = to the current flowing through the capacitor (Ic).
Vc = Vin - R1 * C * dVc/dt - L * d(C * dVc/dt)/dt
Vc'' + R1/L * Vc' + Vc/LC = Vin/L