@ecterrab First, I want to thank you for the explanation and for the help.
If I delete the cosine term, what remains will be exactly the particular solution I am interestred in and also the odetest command returns True.
I can also change the coeff. in front of cosine and what remains will be still a solution of the ODE (why did maple choose exactly coeff 3 if any other coeff. in front of cosine term also leads to a solution of the ODE? Maple just set the arbitrary constant randomly to 3)
If we assume, that " There is nothing in the definition of particular solution that says that it "cannot include terms that, isolated, may also solve the homogeneous ODE." " as you said, then what is the meaning of this function, if it returns not unique solution. (since terms that solve the homogeneous ODE are not uniqe without initial conditions)