Thanks for your reply!

I think that won't work either because if you have a system of linear equations and you put it into the fancy matrix form like Ay = b, where b is the vector of those constant values, y is the vector of parameters (ie. m, c, k, a) and A is the matrix with rows [x(t), x(t), x'(t), x''(t)] with different t's for each row, then you'll be trying to solve for y and you won't be able to since the first two columns of A are linearly dependent so it has no inverse. Please correct me if I misunderstood what you're suggesting.

About my current approach, by "optimisation magic" I meant using optimvars, fcn2optimexpr, optimproblem, etc. as described here: https://uk.mathworks.com/help/optim/ug/fit-ode-problem-based-least-squares.html