I have a problem with homogenous system of two differential equations (fourth order). Only thing which I know is that solutions of characteristic equation (polynomial of 8th order) are complex in the following form
$$s_1=m_1+n_1*i,\; s_2=m_1-n_1*i,\; s_3=-m_1+n_1*i,\; s_4=-m_1-n_1*i,$$ and $$s_5=m_2+n_2*i,\; s_6=m_2-n_2*i,\; s_7=-m_2+n_2*i,\; s_8=-m_2-n_2*i.$$
Is there anyway to obtain analytically solutions for $y_1(x)$ and $y_2(x)$ with 8 unknown constants (constants can be solved using initial conditions). The system is in the following form where $a, b, c$ and $d$ are known constants in the equation.
$$y_1''''(x) + a*y_1''(x) + b*(y_1(x) - y_2(x))=0,$$ $$y_2''''(x) + c*y_2''(x) + d*y_2(x) - b*(y_1(x) - y_2(x))=0.$$