If the determinant of the vectors is not zero, then the vectors are L.I. So simply take the determinant.

with(LinearAlgebra):
A := `<|>`(`<,>`(-1, -3, -6), `<,>`(3, 5, 6), `<,>`(-3, -3, -4));
v,e := Eigenvectors(A);
Determinant(e);
   

       -1

Hence L.I.

You do need to eventually specify an actual u(t) function

restart;

ode:=diff(x(t),t$2)+4*diff(x(t),t)+3*x(t)=u(t);

dsolve({ode,x(0)=0,D(x)(0)=0},x(t));