continuous signals - Linear Constant Coefficient Differential Equations: Zero-Input and Zero-State responses

The solution to a linear constant coefficient differential equation of the form $$\sum_{k = 0}^{N} a_k y^{(k)} (t) = \sum_{k = 0}^{M} b_k x^{(k)} (t)$$ can be written as $y(t) = y_{ZI} (t) + y_{ZS} (t)$ (sum of the zero-input response and the zero-state response) or $y(t) = y_h (t) + y_p (t)$ (the sum of the homogeneous solution and the particular solution). The zero-input response is found by solving the homogeneous equation with initial conditions applied to the solution to the homogeneous equation alone, whereas the zero-state response is found by solving the differential equation with zero initial conditions applied to the solution to the full equation (the sum of the homogeneous solution and the particular solution). However, if we split the solution up to a homogeneous solution and a particular solution (the second decomposition above) we apply initial conditions to the sum of the homogeneous and particular solutions, but not to the homogeneous solution alone first, before adding to it the particular solution. What is the reason for that?

I also wanted to know how this relates to AC steady state analysis in circuit theory. When we do steady state analysis, are we obtaining $y_p (t)$ or $y_{ZS} (t)$? Can you explain why? Is the steady-state response the particular solution or the zero-state response? Is it not one or the other in general?

I would be very grateful if you could explain this in terms of the time-domain description of systems, without resorting to s-plane concepts, poles, etc.