@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)

@ecterrab 

There is description of the function:

"find a particular solution to a nonlinear ODE, or a linear non-homogeneous ODE, without computing its general solution"

But it seems that, sometimes the function cannot filter out the homogeneous solution from the general one. 

@ecterrab Dear Edgardo S. Cheb-Terrab, 

thank you for your answer. As I learned it in my university, general solution of an inhomogeneous ODE consists of two parts: particular solution and solution of the corresponding homogeneous ODE (homogeneous solution). I expected, that the function particularsol gives me the particular solution, but there is the cosine term, which belongs to solution of the corresponding homogeneous ODE.