chapter  9
10 Pages

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.