Ordinary Differential Equations and Boundary Value Problems: Volume I: Advanced Ordinary Differential Equations by John R Graef Johnny Henderson Lingju Kong and Xueyan Sherry Liu

Author:John R Graef, Johnny Henderson, Lingju Kong and Xueyan Sherry Liu
Language: eng
Format: epub
Publisher: World Scientific Publishing Co. Pte. Ltd.
Published: 2018-11-15T00:00:00+00:00

Theorem 5.3. Let x1(t), x2(t), …, xm(t) be solutions of (5.1). Then, if there are constants α1, α2, …, αm and a point t0 ∊ I such that , then it follow that on I.

Consequently, the solution space of (5.1) is an n-dimensional subspace of C (I, ℝn) (or C[I, n]). Furthermore, if x1(t), x2(t), …, xn(t) are solutions of (5.1) such that, for some t0 ∈ I, x1(t0), …, xn(t0) are L.I. in ℝn (or n), then x1(t), …, xn(t) constitute a basis for the solution space of (5.1).

Proof. For the first assertion, assume x1(t), …, xm(t) are solutions of (5.1) and that α1, …, αm are scalars and t0 ∊ I such that . By Theorem 5.2, is a solution of (5.1). Moreover, x(t0)= 0, and so by Corollary 5.1, x(t) ≡ 0 on I.

Assume now that x1(t), …, xn(t) are solutions of (5.1) such that at some t0, x1(t0), …, xn(t0) are L.I. vectors in ℝn. Such solutions exist; e.g., for each 1 ≤ j ≤ n, let xj(t) be the solution of

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