Governing Equation for an electrical system

[mintsnapple]

Homework Statement

Homework Equations

Kirchoff's Voltage/Current Law

The Attempt at a Solution

I first started by summing up the currents at node 1, which is the intersection of 3 wires at the top, and node 2, which is between the capacitor and inductor.
So, at node 1: $$ \sum i = -i(t) + \frac{1}{R}V_1 + C\frac{d}{dt}(V_1 - V_2) = 0$$
At node 2: $$ \sum i = -C\frac{d}{dt}(V_2 - V_1) + \frac{1}{L}\int_{-\infty}^t V_2 dt = 0 $$
I also noted that $$ V_2 = V_L $$

So I have these two equations, but I am not sure how to easily get rid of the V_1 dependence. It's not as simple as just solving for V_1, is it?

 

[NascentOxygen]

Best not introduce more variables than is necessary. I think you'll find that your node 2 does not qualify for the exhaulted designation "node".

Express the voltage acoss the C + L combination in terms of their current, i₂(t)

Express the voltage across R in terms of its current, i(t) - i₂(t)

Equate the two expressions, and rearrange to give i₂(t) in terms of i(t).

Good luck! ☺