  Boost-invariant equations of spin hydrodynamics confined to the first-order
terms in gradients are numerically solved. The spin equation of state, relating
the spin density tensor to the spin chemical potential, is consistently
included in the first order. Depending on its form and the structure of the
spin transport coefficients, we find solutions which are both stable and
unstable within the considered evolution times of 10 fm/c. These findings are
complementary to the recent identification of stable and unstable modes for
perturbed uniform spin systems described by similar hydrodynamic frameworks.
