A place for everything about math

The fact that this comes from a physicall model means that there is a solution, no ?

Good instinct, but nope! There might be some phenomenon your model is neglecting where the neglected phenomenon allows the solution to exist but the model fails to have a unique solution. It also could have a non-unique solution (i.e. many solutions).

Do you have a link to those equivalent system, i can't find it online and anything i can search up on is nice.

So the first move is to simply solve the equation for X'':

X'' = X^-4 f

Unfortunately, for the next technique, it's typically buried in any nonlinear control book. I'll give the technique below. I'm gonna do it for a fourth order system to make the technique clear, but it applies for ODE systems of any size.

Say you have the following fourth order scalar differential equation:

X''''(t) = F(t,X(t),X'(t),X''(t),X'''(t)) Make the following substitutions:

x1 = X
x2 = X'
x3 = X''
x4 = X'''

If you have initial conditions at some time t0, then they must be:

x1(t0) = X(t0)
x2(t0) = X'(t0)
x3(t0) = X''(t0)
x4(t0) = X'''(t0)

(In your problem, t0 = 0, plus your problem is time-invariant so it doesn't matter, but I'm describing the technique here for time-varying problems for generality, where it does matter.)

Differentiate the four equations as follows:

x1' = X'' = x2
x2' = X'' = x3
x3' = X''' = x4
x4' = X'''' = F(t,X(t),X'(t),X''(t),X'''(t)) = F(t,x1,x2,x3,x4)

Stack these into a vector as follows:

[x1,x2,x3,x4]^T^ = [x2,x3,x4,F(t,x1,x2,x3,x4)]^T^

(The ^T^ just means "make it a column vector".) You now have a system of four coupled first-order ODEs that any ODE system solver will solve. By applying the technique described above to the second-order ODE X'' = X^-4 f, you get [x1,x2]^T^ = [x2,x1(t)^-4 f(t)]^T^.

This means that f is drawn from the space of solution ms of a second order homogenous liner differential equation (from an RLC circuit with no generator).

Got it, so the forcing is a sum of exponentials. So your input space is a subspace of the infinitely differentiable functions (except possibly at 0 if you choose to model the switching-on of the circuit as a step function). But I'm a little suspicious of the X^4^ term? Like I'm not a "magnets guy" although I've taken electromagnetic fields, but I don't think that's the type of nonlinearity I would expect to see (in particular , the fact that it's fourth power). I.e. if you derived this model yourself, double-check the derivation.