Eigenvalue problems of the form $x'' = -\lambda f(x) + \mu g(x),$ $\quad (i),$ $x(0) = 0, \; x(1) = 0,$ $\quad (ii)$ are considered. We are looking for $(\lambda, \mu)$ such that the problem $(i), (ii)$ has a nontrivial solution. This problem generalizes the famous Fu\v{c}ik problem for piece-wise linear equations. In order to show that nonlinear Fu\v{c}ik spectra may differ essentially from the classical ones, we consider functions $f(x)$ and $g(x)$ such that they are piece-wise linear and the first zero functions $t_1$ and $\tau_1$ can be computed explicitly. Then it is possible to construct explicitly the respective Fu\v{c}ik like spectra.