# Find the differential equation for Vo (RLC circuit)

Find the differential equation for Vo

My attempt at a solution:
Node V1:

$$\frac{V_1-V_{in}}{R_1} + \frac{1}{L}\int_{0}^{t}(V_1-V_2) = 0$$

Node V2:

$$\frac{1}{L}\int_{0}^{t}(V_2-V_1)+C*\dot{V_2}+\frac{V_2}{R_2}=0$$

Am I doing this correctly? How would I solve for V2? V2 is equal to Vo, correct? If the input (Vin) is a square wave, how would I find the transient and forced responses (assuming I'm given numerical values for R1, R2, L1, C1)?

• To be complete you have missed the node between Vin and V1: (V0 - V1)/R1 = I_vin, and V0 = Vin Oct 17 '17 at 14:12

Your equations are correct. Differentiate node $\small V_2$ equation and obtain an expression for $\small V_1$. Substitute this expression into the node $\small V_1$ equation. This gives a 2nd order equation in $\small V_2$.
$$\small\ddot{V_2} +\left(\dfrac{1}{R_2C}+\dfrac{R_1}{L}\right)\dot{V_2}+\left(\dfrac{1}{LC}+\dfrac{R_1}{R_2 LC} \right)V_2=\dfrac{V_{in}}{LC}$$
• I ended up with this. Is it correct? $$\ddot{V_2}+\frac{\dot{V_2}}{L}(\frac{R_1}{R_2}+1)+\frac{V_2}{LC}(\frac{R_1}{R_2}+1)=\frac{V_{in}R_1}{R_2LC}$$