Building an nth order ODE in Maple (or Matlab)

Question

The question is simple: given a system of ODEs, how can one construct the equivalent nth order ODE in Maple? In my case I have

$$ \begin{cases} y''(t)+x'(t)+x(t)=f(t)\\ y''(t)+z''(t)+z'(t)+z(t)=0\\ x(t)=y(t)+z(t) \end{cases} $$

and I would like to get a 4th order ODE in x(t) with the same solutions as the system above. Is there any Maple command to do this? (or maybe Matlab?)

Thanks for the help!

Answer 1

des:= {diff(y(t),t$2)+ diff(x(t),t) + x(t) = f(t), 
    diff(y(t),t$2) + diff(z(t),t$2) + diff(z(t),t) + z(t) = 0,
    x(t) = y(t) + z(t)};
DEtools[casesplit](des,[z(t),y(t),x(t)],arbitrary={f});

$$ \eqalign{\bigg[ z \left( t \right) & = x \left( t \right) +{\frac {{\rm d}^{3}}{{\rm d}{t }^{3}}}x \left( t \right) +{\frac {\rm d}{{\rm d}t}}x \left( t \right) -f \left( t \right) \cr y \left( t \right) &=-{\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}x \left( t \right) -{\frac {\rm d}{{\rm d}t}}x \left( t \right) +f \left( t \right) \cr {\frac {{\rm d}^{4}}{{\rm d}{t}^{4}}}x \left( t \right) &\left.= -2\,{\frac { {\rm d}^{2}}{{\rm d}{t}^{2}}}x \left( t \right) -2\,{\frac {\rm d}{ {\rm d}t}}x \left( t \right) +{\frac {\rm d}{{\rm d}t}}f \left( t \right) -x \left( t \right) -{\frac {{\rm d}^{3}}{{\rm d}{t}^{3}}}x \left( t \right) +f \left( t \right) \right] \cr \&where \;[]} $$