a particular solution of the forced equation

Once the structure theorem tells you the answer is y = y_c + y_p, your real job is to produce ONE function y_p that makes the forced equation true — any single solution that includes no arbitrary constants. That single working solution is the particular solution. It is the part of the answer that responds to the forcing rather than to the initial conditions.

Concretely, y_p is any function satisfying y'' + p(x) y' + q(x) y = g(x) on its own (no free constants attached). You typically find it by undetermined coefficients (guess a shape matching g, solve for the coefficients) when g is a polynomial, exponential, sine, cosine, or product of these and the coefficients are constant; otherwise you use variation of parameters, which constructs y_p from the homogeneous solutions and an integral of g. Either way you only need ONE such function — you are not trying to find all of them.

Because the structure theorem absorbs any leftover freedom into y_c, the particular solution is not unique: add any homogeneous solution to a valid y_p and you get another valid y_p. That is why two correct write-ups can differ. Physically, in a stable system y_p is the steady-state (the lasting, forced behaviour), while y_c is the transient. Keep them distinct: y_p alone usually will NOT satisfy the initial conditions — those are met only after you add y_c and choose its constants.