A place for everything about math

Hmm...this feels like a Lyapunov-type problem, but there's some issues. The vector function [x2,x1^-4 f] has no equilibrium, except when f = 0. So assume that f = 0. Then, the system is [x1',x2'] = [x2,0] = [0 1;0 0][x1,x2]. I believe that the zero solution satisfies your original differential equation. You would then need to find a Lyapunov function that guarantees the system is globally asymptotically stable around the zero solution. (Make sure to check for "globally asymptotically" in the stability theorem, e.g. in Theorem 3.2 of Haddad and Chellaboina (LibGen). The "asymptotic" part is what gives you the steady-state result you're looking for.) By taking the second component, I would then conclude that the limit of X' is the limit of x2, which would be 0.

But that's assuming zero input and initial state, and Lyapunov functions can be hard to find. Linearizing fails; you would have to guess a Lyapunov function.

Honestly, from an engineering perspective, I think you should just simulate the system. In particular, instead of leaving f(t) as a function, I suggest plugging in the second-order ODE and numerically solving the resulting composite fourth-order ODE system for your RLC values and some deviations from those values. Simulating it might give you an ansatz (educated guess) on if or where the system has equilibria.