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I've got this system of differential equations $$\begin{cases} x''=f(t,x,x',y,y')\\ y''=g(t,x,x',y,y') \end{cases}$$

Where $f$ and $g$ are given and known functions and $x=x(t), y=y(t)$

Is there any algorithm or method to bring this system to canonical(normal) form?

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    What canonical normal form are you refering here? Of course, you can transform this system of *two* equations of the *second order* to the system of *four* equations of the *first order*. It's very easy to do, but I'm not sure that this is what you mean here. – 2017-01-17
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    Evgeny that's exactly what I wanna do, how can I do that? – 2017-01-17
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    The equivalent system is $$ \left \lbrace \begin{array}{ccc} \dot{x} &=& u \\ \dot{u} &=& f(t, x, u, y, w) \\ \dot{y} &=& w \\ \dot{w} &=& g(t, x, u, y, w) \end{array} \right . $$ – 2017-01-17

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