How to solve the following equation by Green's function Method?
$$x^{(4)}_1(t)+(b_1+b_3)x''_1(t)+(b_1b_3-b_4b_2)x_1(t)={g}''_1(t)+b_3g_1(t)+b_2g_2(t)$$ with initial conditions: $$x_1(0)=m_1$$ $$x_1'(0)=m_2$$ $$x_2(0)=m_3$$ $$x_2'(0)=m_4$$ where $b_j,m_j$ are constant for $i=1,2,3,4$
and $x_2=(x_1''+b_1\,x_1-g_1)/b_2$
In fact, I seek the solutions $x_1(t)$ and $x_2(t)$ which are depended on $g_1(t)$ and $g_2(t)$.