Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as things that vanish "at infinity". First, let me mention that the relative cohomology is relative de Rham cohomology (the mapping cone of the morphism between the complexes of differential forms), which I think kill some pathologies regarding bad pairs (because it's the homotopy cockerel and not the ordinary cokernel!).
Let me first say that I do know how to prove $H (E, E^0) \cong H_{cv} (E)$ by applying the comparison isomorphism for de Rham and singular cohomology and using the Thom isomorphism on each side (the differentiable in one side and the topological on the other), however I can't write this isomorphism explicitly.
To achieve this, I have three suggestions (maybe they will fail, I don't know). Let focus on $H (D, D-\{x\}) \cong H_c (D)$ first. I think I can glue everything to the vector bundle case using partition of unity (as in the construction of the angle form in Bott and Tu).
1) Let $D_r$ denotes the disk of radius $r > 0$. My first suggestion is to establish an isomorphism $H (D, D-\{x\}) \cong H_c (D, D- D_r)$ for each $r$ and then take the limit $r \rightarrow \infty$.
2) Let $f : D - \{x\} \rightarrow D$ be the map with image $x$. There's an homotopy between the inclusion $D - \{x\} \hookrightarrow D$ and $f$. This induces an isomorphism $H (f) \cong H (D, D-\{x\})$. Now, since $f$ factors through the inclusion of a point, I think that $H(f) \cong H (D, \{x\})$.(I'm not sure how cohomology works for maps $f$ that are not inclusion. Apparently this reduces to the property of the cohomology commuting with cofibers (see my remark about the homotopy cockerel in the first paragraph))
3) This is the more elementar approach. Let $\omega \in \Omega_c^i (D)$ be closed and $\eta \in \Omega^i (D-\{x\})$ be a form such that $d\eta = \omega|_{D-\{x\}} $. Define the map $[\omega] \mapsto [(\omega, \eta)] \in H^i (D, D-\{x\})$. I couldn't prove this is an isomorphism. More precisely, I couldn't find an element in each cohomology class in $H^i (D, D-\{x\})$ such that the first coordinate have compact support (this seems to be some trick argument using bump functions).
Now the last step after proving one of the three above is to glue explicitly using partition of unity to an isomorphism $H (E, E^0) \cong H_{cv} (E)$
Any ideas are welcome. Thanks in advance.