I'm trying to write and solve a differential equation for this simple RLC notch filter and I'm not sure what I'm doing wrong.
simulate this circuit – Schematic created using CircuitLab
So first, I apply KVL to (1) the loop source-resistor-capacitor-source and (2) the loop source-resistor-capacitor-source to get what's essentially an RC and RL circuit:
\$ RC\frac{dI_R}{dt} + I_c = C\frac{dV}{dt}\$
\$ L\frac{d^2I_L}{dt^2} + I_R = V\$
Adding these, I get
\$ L\frac{d^2I_L}{dt^2} + 2R\frac{dI_r}{dt} + \frac{I_C}{C} = 2\frac{dV}{dt} \$
Using KCL I know that
\$ I_R = I_C + I_L \$
Using impedances, I get
\$ Z_C = \frac{1}{j\omega C} \$ and \$ Z_L = j\omega L \$
Based on the current divider equation, I have
\$ I_L = \frac{Z_C}{Z_L}I_R \$ and \$I_C \frac{Z_L}{Z_C}I_R \$
\$ I_C = -I_R \omega^2 LC \$
\$ I_L = -\frac{I_R}{\omega^2 LC} \$
If I put this into the differential equation, I get
\$ \frac{d^2I_R}{dt^2} -2RC \omega^2\frac{dI_R}{dt} + \omega^4LCI_R = -2\frac{dV}{dt} \$
If I try to solve this, I get positive exponentials, which I know are wrong since current won't increase without bound. I'm not sure how else to go about this. Any help is appreciated. Thanks!