Math Help - Parametric differential equation

Hello everybody,
i'm really stuck with this and i hope somebody can help me out.

I have this differential equation:

sadly there is no further explaination on what i should do with the parameter, so here is how i got through:
i wrote the omogeneous equation:

then tried to find its general solution by writing the characteristic equation:


which has the following zeroes:
real:

and
complex:

Which leads to the general solution of the omogeneous equation.
Now, on to the parameter; due to the fact that the coefficient, correct me if i'm wrong, changes the way my particular solution of the complete equation is wrote, should i study the cases where equals the characteristic equation zeroes? And if this is the right procedure, when i study the case in which:
or
my solution will have the complex exponent?
Such as: ?

Hope this is a clear enough question, excuse me for my poor english

and thanks in advance,
SkyWolf.

No problem about a particular solution. According to a well known theorem, if is not a root of the characteristic equation a particular solution has the form . If is a root of the characteristic equation of multiplicity , a particular solution has the form . Of course if you needn't consider the roots .

Thanks a lot man.