solution forms for repeated roots

The characteristic polynomial promises n roots, and you want n independent solutions to match. But what if a root shows up more than once — say r is a double or triple root? Then e^(rx) alone gives you only ONE solution where the count demands more, and the simple recipe runs short. Repeated-root solution forms are the rule that supplies the missing solutions, and the trick is to bring in factors of x.

The pattern is clean. If r is a real root of multiplicity m (it appears m times in the factored characteristic polynomial), then it contributes m linearly independent solutions: e^(rx), x e^(rx), x^2 e^(rx), ..., x^(m-1) e^(rx). So a double root r gives e^(rx) and x e^(rx); a triple root gives e^(rx), x e^(rx), x^2 e^(rx). For a repeated COMPLEX pair a +/- bi of multiplicity m, the same idea doubles up on the sine-cosine pair: each of e^(ax) cos(bx) and e^(ax) sin(bx) gets multiplied by 1, x, x^2, ..., x^(m-1). Tallying every root with its multiplicity always lands you exactly n solutions, and because they are independent they form a fundamental set.

Why does multiplying by x work? Intuitively, two equal roots are the limit of two nearby distinct roots r and r + h: the combination (e^((r+h)x) - e^(rx))/h is a solution, and as h shrinks to zero it tends to x e^(rx). So the x factor is the fingerprint of a collision of roots. This shows up physically as critical damping at second order and its higher-order analogues. The common mistake to avoid: never just repeat e^(rx) twice — repeating the same function adds nothing new, leaves the Wronskian zero, and fails to span the space.