Coupled second order differential equation

I need to solve a coupled system of linear second order differential equations of the form:

a1x′′+b1x′+c1x+d1y=0a1x″+b1x′+c1x+d1y=0

a2y′′+b2y′+c2y+d2x=0a2y″+b2y′+c2y+d2x=0

Initial conditions are :

y(0)=y0+y′(0), y′(H)=0 ,x(0)=x0+x′(0) ,x′(H)=0.

y(0)=y0+y′(0) ,y′(H)=0,x(0)=x0+x′(0),x′(H)=0.

x and y are displacement in height.

This system shows the balance in chemical system between the concentration of gas phase and liquid phase.

What method do you suggest for solving this system? Any suggestion will be appreciated!

Thanks!