# Can we solve the ODE system with Green Function Methods?

Can we solve the following ODE system by Green Functions Method? Could you give some hints?

$\ddot{x}_1(t)+a_1x_1(t)-a_2x_2(t)=f_1(t)$,

$\ddot{x}_2(t)+a_3x_2(t)-a_4x_1(t)=f_2(t)$,

where $a_1,a_2,a_3,a_4$ are constants.

The given conditions are as follows: ${x}_1(0)=k,{x}_2(0)=l,\dot{x}_1(0)=m,\dot{x}_2(0)=n$ where $k, l, m, n$ are constants.
($f_1(t),f_2(t)$ are ungiven functions.)

(You can find the transformed first order differential eq's. system of the second order system)

• I would write the second order system as a first order system, then use first order Green's function methods. In either case, finding the Green's function is the main problem, as you can simply integrate to get the solution. – AlexanderJ93 Apr 24 '17 at 7:14
• If we define $z_1=x_1,z_2=\dot{x}_1, z_3=x_2,z_4=\dot{x}_3$, we get the following first order ode system: $$\dot{z}_1(t)=z_2(t)$$ $$\dot{z}_2(t)=-a_1z_1(t)+a_2z_3(t)+f_1(t)$$ $$\dot{z}_3(t)=z_4(t)$$ $$\dot{z}_4(t)=-a_3z_3(t)+a_4z_1(t)+f_2(t)$$ could you help the rest ? – HD239 Apr 24 '17 at 7:34