Does the method of undetermined coefficients and varation of parameters give you the same particular solution?

Answer 1

Yes, it has to. The only difference between the two methods is that the undetermined coefficients method is generally faster, but applies only to a restricted class of rhs functions. The variation of parameters method is usually longer, but can be used on any right-hand side function whatsoever.

Answer 2

There is no reason that both methods should give you the same particular solution