By Earl A. Coddington

"Written in an admirably cleancut and cost-efficient style." — *Mathematical Review*. a radical, systematic first direction in undemanding differential equations for undergraduates in arithmetic and technological know-how, requiring merely uncomplicated calculus for a history, and together with many routines designed to strengthen students' process in fixing equations. With difficulties and solutions. Index.

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In particular, the real Jordan matrix for complex roots would have been simply a pair of complex entries in the diagonal matrix J 0 above. 21 the matrix p- 1 AP will often be complex. 3 will not hold for y' = J y. 10) into a system y' = Jy which is a set of entirely independent systems of linear differential equations. The PMS reflects this as we see that e" - riot e'" [0 0] e J qt • In order to more fully utilize the representation of Z(t, 0) as eA I and to display the solutions of y' = Jy we present the following result.

14), 0: = 0, P< o. 1. linear Equations 46 Fig. 14), IX < 0, f3 < 0. Fig. 14), IX < 0, f3 > 0. 13) are ellipses. The zero solution is stable, but not asymptotically stable. When ex. < 0, the portraits are given in Figs. 13. In these cases, the origin is called a focus. It is asymptotically stable. 6. 1) x' = A(t)x and a scalar equation, say x' = a(t)x. Perhaps the most notable similarities are seen in the Jacobi identity and in the variation of parameters formula. 10) and the scalar equation are formally the same.

1) is Liapunov stable if and only if Z(t, to) is bounded. It is understood that Z is bounded if each of its elements is bounded. first suppose that Z(t, to) is bounded, say IZ(t, to)1 :;;; M for to :;;; t < 00. t 1 ~ to and G> 0 be given. We must find {) > 0 such that IXol < {) and t 1 imply Ix(t, t 1, xo)! < e. 2 x = m. Thus we take f> = e/Mm. 4. That is, assume show that Z(t, to) is bounded. 1 we have Z(t, to) bounded and hence x = 0 is Liapunov stable. Furthermore, show that f> may be chosen independent of t 1 in this case.