eigenvalue method
To solve a single constant-coefficient equation you guess y = e^(r t). To solve a system x' = A x you make the matching guess: x = v e^(lambda t), a fixed direction vector v riding a single exponential. The eigenvalue method is the systematic way of finding which directions and which growth rates the matrix A actually permits.
Substituting x = v e^(lambda t) into x' = A x gives lambda v e^(lambda t) = A v e^(lambda t), so A v = lambda v: v must be an eigenvector of A and lambda its eigenvalue. You find them by solving det(A - lambda I) = 0 for the eigenvalues and then the eigenvectors. Each eigenpair gives one solution v e^(lambda t); superposing n independent ones gives the general solution. The three root cases mirror the single-equation story: real distinct eigenvalues give pure exponential modes; complex eigenvalues alpha plus-or-minus i beta give spiraling oscillations e^(alpha t) times sinusoids; a repeated eigenvalue with too few eigenvectors needs an extra generalized-eigenvector solution carrying a factor of t.
The eigenvalue method is the bridge between differential equations and linear algebra, and it is why eigenvalues are so central to applied science. Each eigenvalue is a natural rate; each eigenvector is a natural mode — a special initial state in which the whole system evolves by a single exponential without changing shape. Normal modes of vibration, principal axes, and the stability of equilibria all come straight from this picture: the signs of the real parts of the eigenvalues decide whether the system decays to rest or runs away.
For A = [1, 1; 4, 1], det(A - lambda I) = (1-lambda)^2 - 4 = 0 gives lambda = 3 and lambda = -1, with eigenvectors (1, 2) and (1, -2). So x = c_1 (1, 2) e^(3t) + c_2 (1, -2) e^(-t).
Each eigenpair contributes one mode v e^(lambda t); superposing them gives the general solution.
A repeated eigenvalue does not automatically need a 't' term — only when it is 'defective', i.e. has fewer independent eigenvectors than its multiplicity. If it still supplies enough eigenvectors, ordinary exponential modes suffice.