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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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The solution is attached below in two files. the files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe ...

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.