Solution

Mathematically, even though the particular solution (as a result of external forcing) satisfies the equation of motion, it cannot satisfies initial conditions (except in a very specific situation). That is why the general solution must contain the complementary solution (as a result of restoring force) as well.

Physically, as soon as the external forcing displaces the object from its equilibrium, the restoring force comes into play. That is why the total motion is the sum of free and forced oscillations.