Why do forced vibration solutions need both complementary and particular parts?

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chandran
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in deriving the solution for displacement in forced vibration theory is derived

for complimentary solution and particular solution.

Can anyone throw some light on the above solution. Why there are two solutions?
 

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Because the complimentary or fundamental solutions are for homogenous linear ODEs, while the particular solution is for non-homogenous linear ODEs, but because of the linearity of the differential operator the combination of both solutions will be the complete set of solutions for the ODE.