The characteristic solution follows from solving the characteristic equation,

so that

A guess for the particular solution may be

, but this is already contained within the characteristic solution. We require a set of linearly independent solutions, so we can look to

which has second derivative

Substituting into the ODE, you have



Therefore the particular solution is

Note that you could have made a more precise guess of

but, of course, any solution of the form

is already accounted for within

.