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Define "disjoint" in this context.Mark44 said:With regard to ##y'' + 9y = \frac 1 2 \cos(3x)##:Sure, this is a particular solution, one that can be decomposed into ##\frac x {12}\sin(3x)## and ##\frac 1{12}\sin(3x)##. The former is a solution to the nonhomogeneous DE above, and the latter is a solution of the homogeneous equation.

My whole argument here is why go out of one's way to devise a particular solution that isn't disjoint from the homogeneous problem's solution set?