Download A Short Course in Ordinary Differential Equations by Qingkai Kong PDF

By Qingkai Kong

This textual content is a rigorous therapy of the fundamental qualitative thought of normal differential equations, firstly graduate point. Designed as a versatile one-semester direction yet delivering adequate fabric for 2 semesters, a quick path covers middle themes equivalent to preliminary price difficulties, linear differential equations, Lyapunov balance, dynamical platforms and the Poincaré—Bendixson theorem, and bifurcation idea, and second-order subject matters together with oscillation conception, boundary worth difficulties, and Sturm—Liouville difficulties. The presentation is obvious and easy-to-understand, with figures and copious examples illustrating the which means of and motivation in the back of definitions, hypotheses, and basic theorems. A thoughtfully conceived choice of routines including solutions and tricks strengthen the reader's realizing of the cloth. necessities are constrained to complex calculus and the uncomplicated thought of differential equations and linear algebra, making the textual content compatible for senior undergraduates as well.

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However, we should note that the solutions of Eq. (H-p) are not necessarily periodic. First we introduce certain properties of Eq. (H-p) which are not satisfied by the general homogeneous linear equation (H). 1. If x(t) is a solution of Eq. (H-p), then x(t + kω) is also a solution of Eq. (H-p) for each k ∈ Z := {0, ±1, . ± 2, . . }. Proof. Since x(t) is a solution of Eq. (H-p), we have x (t) = A(t)x(t) for t ∈ R. Then for t ∈ R and k ∈ Z, [x(t + kω)] = x (t + kω) = A(t + kω)x(t + kω) = A(t)x(t + kω).

Then φ(t) can be extended to a maximal interval of existence (α, β) in a countably infinite number of steps; and (t, φ(t)) → ∂D, the boundary of D, as t → α+ or t → β−. 4. MAXIMAL INTERVAL OF EXISTENCE 19 Furthermore, if f is locally Lipschitz in x on D, then the extension is unique. 1. 1, by (t, φ(t)) → ∂D as t → α+ or t → β− we mean that for any compact set G ⊂ D, (t, φ(t)) ∈ G when t is sufficiently close to α or β. 1. For k = 1, 2, . . , define (t, x) ∈ D : |t − t0 | < k, |x − x0 | < k, and d((t, x), ∂D) > Dk = 1 k ¯ k is the closure of Dk , where d((t, x), ∂D) means the distance from the and D point (t, x) to ∂D.

In fact, by definition, ∞ X(t) = k=0 Ak (t − t0 )k , k! t ∈ R. 4. HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 43 For any r > 0 and t ∈ [t0 − r, t0 + r], (|A|r)k Ak (t − t0 )k ≤ . k! k! ∞ Since the scalar series k=0 |Ar|k /k! is convergent, the matrix-valued funck k tion series ∞ k=0 A (t − t0 ) /k! is uniformly convergent on [t0 − r, t0 + r]. Consequently, for any t ∈ (t0 − r, t0 + r), ∞ X (t) = k=0 ∞ = A k=1 ∞ Ak (t − t0 )k k! A k=0 (t − t0 ) (k − 1)! k−1 Ak (t − t0 )k k! = ∞ k−1 =A k=0 ∞ = k=1 Ak (t − t0 )k−1 (k − 1)!

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