a homogeneous linear system
Before tackling a system that is being pushed by some outside input, you study the system left alone — switch off every external source and watch how it moves on its own. That undriven, self-contained version is the homogeneous linear system, and it turns out to be the heart of the whole problem.
It is the linear system with no input term: x' = A(t) x, or in the constant-coefficient case the clean x' = A x. With g(t) gone, the zero vector x = 0 is always a solution (start at rest, stay at rest forever) — the equilibrium of the system. The deeper fact is superposition: if x_a(t) and x_b(t) are both solutions, then so is any combination c1 x_a(t) + c2 x_b(t). Solutions thus form a vector space, and for an n-dimensional system that space is exactly n-dimensional. So solving a homogeneous system means finding n independent solutions; every other solution is a linear combination of those.
Homogeneous systems matter twice over. First, the constant-coefficient case x' = A x is solved completely by the eigenvalues and eigenvectors of A — this is where straight-line solutions, complex oscillations, and generalized eigenvectors all live. Second, the homogeneous solution is half of every forced problem: the general solution of x' = A x + g(t) is one particular solution plus the entire homogeneous family. Master the homogeneous system and the forced one is a short further step.
x' = A x with A = [0, 1; -1, 0] describes pure rotation: x1' = x2, x2' = -x1. Two independent solutions are (cos t, -sin t) and (sin t, cos t); every solution is c1 times the first plus c2 times the second.
Solutions of a homogeneous system close under scaling and addition — that is superposition at work.
Superposition is a privilege of linearity alone. For a nonhomogeneous system, adding two solutions does NOT give a solution — the input term g(t) would be counted twice.