Hey, i got a few questions :( sorry

1 - I haven't come across the term "Bump Up" in all my notes and texts, can you explain?

2 -WELL I've tried to do it again, here's what I've got. PLEASE tell me where I've gone WRONG :(

The characteristic equation:

The roots to this equation:

and

therefore the complementary function:

Mr F says: Correct in spirit I suppose. But shouldn't it be ....
so my

Mr F says: This part is correct. But there's another part that you've forgotten to include. See main reply below.
therefore my general solution is

Thanks again :)