Let $M$ be a non-compact differential manifold. It is true that in general $H^q_c(M) \neq H^q(M)$, where $H^q_c$ is the de Rham's cohomolgy with compact support group and $H^q$ is the usual de Rham's cohomology group.
We have just begun the subject, so I don't have much confidence with it. I wanted to ask: is it true that for any non-compact $M$ there exists a $q$ for which $H^q_c (M) \neq H^q(M)$? Or are there any examples of non-compact manifolds for which the two cohomologies are the same for all $q$?
EDIT: ok, from the comments I gathered that if such an example exists it must be non-orientable (a reference to a proof would be nice, even though I think the resul is quite non-elementary). The question still remains open though (that's my main reason to editing: I think this question didn't get enough attention).