This is a homework question so please don't give the full answer, just an approach will do.
Question: Let $T=D_u\cup\{c_i<c_{i+1}\mid i\in\mathbb N\}$ where $D_u$ is the theory of dense linear orders without end points.
(a) Show that $T$ has three non-isomorphic models of size $\aleph_0$, (you may use that $D_u$ is $\omega$-categorical).
(b) Show that $T$ is complete.
(a) was easy enough but I'm stuck on (b).
Now my first attempt was to simply say "$T$ is consistent because it is satisfiable, $D_u$ is complete by Vaught's test, so $T$ is complete." That's missing a couple of steps but you get the idea.
Then I realised that $D_u$ isn't actually complete here, because the language is $\{<\}\cup\{c_i\mid i\in\mathbb N\}$, which means we have sentences like $c_1<c_2$.
So because $D_u$ is complete with respect to $\{<\}$, the only problem sentences can be those that involve constants, but I don't see where to go from there.
I think for part a) you also want that "non-isomorphic models of size $\aleph_{0}$". For b) a back and forth won't work (since the models are not isomorphic). What about using EF games?