I am trying to find the expressions $V_k$ and $\psi_k$ of this equation:
The source: On the nonlinear electromagnetic coupling between a coil and an oscillating magnet
link: http://iopscience.iop.org/article/10.1088/0022-3727/43/29/295005/meta
N, $\psi$ and $\lambda$ are all constants r,x and z are just geometrical parameters.