Appendix: Basic background mathematics
In case a = 0 and all a0, a1, . . . , an−1 are constant coeﬃcients, the general solution of Eq. (9.3) can be written explicitly. It is readily seen that x¯(t) = exp(λt) is a solution of (9.3) provided that λ is a solution of the following algebraic equation:
λn + an−1λn−1 + . . . + a1λ + a0 = 0. (9.4)
Solutions of Eq. (9.4) are called the eigenvalues of the ODE (9.3). The form of the general solution depends on whether Eq. (9.4) has multiple
solutions or all the eigenvalues are diﬀerent. Let us consider these two cases separately.