Following the conventions of the DynamicSystems package, u1(t) is your input, so you need to tell dsolve to solve for the others (x1(t), x2(t), x3(t) are states, and y1(t) is the output):
sys := [diff(x1(t), t) = x2(t), diff(x2(t), t) = -x3(t)/3, diff(x3(t), t) = 6*x2(t) - 3*x3(t) - 3*u1(t), y1(t) = -6*x1(t) + x2(t)]:
fcns := [x1(t),x2(t),x3(t),y1(t)]:
sol := dsolve(sys,fcns);
odetest(sol,sys); # optional check
So you have 4 equations and 4 unknown functions. Integration constants _C1, _C2, _C3, _C4 will be determined when you supply initial conditions for fcns. The integrals in those solutions can be computed once u1(t) is specified.
You could also swap the roles of u1(t) and y1(t), of course.