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Method of Undermined coefficients

I am working on the differential equation
(dx^2)/(dt^2) + dx/dt + x =sin (ωt)
I have found the general solution of m^2+m+1=0 which is
I am looking for a particular integral that satisfies the differential equation so as to obtain the general solution
I am finding great difficulty in finding the particular integral xp(t) through the method of undetermined coefficients because i seem to not be able to guess the form of the solution properly probably to a failure of basic maths ability

I would appreciate a few words on the thought process to arrive at how we make a guess of the form of the solution
so far (and i am not sure this is right) I have
and differentiate twice
I then substitute into left hand side
we get
which gives
so by now i am completely lost
please help me to understand how to arrive at the general solution

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Solution Summary

In the first three pages I solve the homogeneous equation and get homogeneous solution.

In pages 4-6 I solve for the particular solution in the method of undetermined coefficients, including a MAPLE verification of the solution.

In pages 7-10 I show how to use the method of variation of parameters. I show how to get the integrals (but I do not solve them) for the particular solution.