a nonhomogeneous system
A homogeneous system x' = A x describes things left alone — temperatures relaxing, populations interacting, currents settling — with no outside push. But real systems are usually driven: a heater pumps in heat, a generator supplies voltage, immigration adds to a population. A nonhomogeneous system is a linear system with such an external input added on: x' = A x + g(t), where g(t) is a given forcing vector, often varying in time.
Its solution has a clean two-part structure that mirrors the single-equation case exactly. The general solution is x(t) = x_h(t) + x_p(t): a complementary part x_h that is the general solution of the homogeneous system x' = A x (built from e^(At) and an arbitrary constant vector), plus any one particular solution x_p that handles the forcing g(t). The homogeneous part carries the system's own natural behavior and the freedom to match any initial condition; the particular part is the response specifically to the input. Superposition guarantees this split works, and it also means that if the forcing is a sum of pieces, you can solve for each piece's response separately and add the results.
The natural decay or growth in x_h is the transient — for a stable A it dies away, leaving the forced response to dominate the long run, exactly as a driven oscillator settles into the rhythm of its driver. Finding x_p is the real work, and the all-purpose tool for systems is variation of parameters, which produces the Duhamel convolution integral against e^(At). For special forcing (constants, polynomials, exponentials, sinusoids) an undetermined-coefficients guess can be quicker.
Two tanks exchanging brine settle to fixed concentrations on their own (the homogeneous part), but if you keep pouring fresh salt into the first tank, g(t) is that steady input. The full salt levels are the natural mixing transient x_h, which fades, plus a particular response x_p that tracks the steady inflow — after a while only x_p remains, the new equilibrium set by the forcing.
General solution = complementary (the system's own behavior) + particular (its response to the forcing); the same homogeneous-plus-particular split as a single equation.
Adding a particular solution does not change the homogeneous family — the arbitrary constants live entirely in x_h, so you impose the initial condition only after assembling x_h + x_p, never on x_p alone.