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How to solve the system of two second order differential equations?:

$$x''(t)=f(x',x,y',y,t),\qquad y''(t)=g(x',x,y',y,t)$$

Is there any numerical method, algorithm?

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The usual trick is to write this system as a first-order one by setting $$U=(U_1,U_2,U_3,U_4)=(X,Y,X',Y')$$ and getting $$U'=(U_3,U_4,f(U_3,U_1,U_4,U_2,t),g(U_3,U_1,U_4,U_2,t)) = F(U,t)$$ Then you can use any of the several goods numerical methods, such as Runge–Kutta.

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I did it, but could not find Runge-Kutta for 4 first order diff equations. Can you help me please to find such an algorithm? – Grigori May 27 '11 at 15:45
    
@Grigori, just interpret RK as a vector method. – lhf May 27 '11 at 15:51
    
What it means? In handbook of Abramovits there is RK method only for system of 2 first order diff equations. – Grigori May 27 '11 at 15:56

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