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These are replies submitted by arcade

@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)



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. 

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