Question: How to eliminate the derivative of u(t) after succeed to eliminate the derivative of lambda

Real application running a very long time in elimination step

z1 := Diff(x1(t),t) = x2(t);

z2 := Diff(x2(t),t) = x3(t);

z3 := Diff(x3(t),t) = x4(t);

z4 := Diff(x4(t),t) = -1*(1.0000*x4(t)-.5109*x3(t)-0.595e-1*x2(t)+3.1086*10^(-15)*(Diff(u(t), t, t, t))-(Diff(u(t), t, t))+.5109*(Diff(u(t), t))+0.595e-1*u(t));

with(Involutive):

with(OreModules):

Alg := DefineOreAlgebra(diff=[D,t], polynom=[t], comm=[a1,a2,a3,u]):

DD := evalm([[D, -1, 0, 0, 0],[0, D, -1, 0, 0],[0, 0, D, -1, 0],[0, -0.0595, -0.5109, D+1, (3.1086*10^(-15))*D^3-D^2+0.5109*D+0.0595]]);

DDtilde := Involution(DD,Alg);

ApplyMatrix(DDtilde, [lambda1(t),lambda2(t),lambda3(t),lambda4(t)],Alg)=evalm([[miu1(t)],[miu2(t)],[miu3(t)],[miu4(t)],[miu(t)]]);

E := Elimination(DDtilde, [lambda1,lambda2,lambda3,lambda4],[miu1,miu2,miu3,miu4,miu],Alg);