solution of Coupled second-order differential equations

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

$a_1x''+ b_1x' + c_1y'' + d_1y' =0$

$a_2x'' + b_2x' + c_2y'' + d_2y' + e_2y = 0$

initial conditions are :

$y(0)=y_0, y'(0)=x(0)=x'(0)=0.$

x and y are displacement in time.

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

Thanks!

• Try writing it in matrix form and computing the matrix exponential. Jan 20 '16 at 18:29
• @Moo to explain more detail the system I should say that the system is a motion equation for coupled (sway and roll; in hydrodynamic), there are solutions for diagonal mass matrix when M12=M21=0, but in my case all paramaters of mass matrix are non-zero value (m11, m12, m21 and m22). Jan 21 '16 at 12:25

In case of linear ODE with constant coefficients, it is known that the general solution is a linear combination of exponential functions on the form $k\,e^{r\,t}$ where the various values of $r$ , real or complex, are the roots of a polynomial equation. So, a method to solve the system is to presume of the form of particular solutions and determine the parameters of them, as done below : 