Solve the following differential equation:

so starting off with the auxiliary equation I have:

where the only available solution is

so I would have:

working on the particular solution I let

substituting for

I come up

Mr F says: . Therefore .
(this is the part that I get stuck on) Since

is a polynomial of third degree I would have:

where

Mr F says: . Note the blue bit. This will effect all the calculations that follow.
substituting I get:

, then

so

and

where I don't think that I'm approaching it correctly since i have 4 unknowns.

any help would be greatly appreciated.