By Qingkai Kong

This article is a rigorous therapy of the fundamental qualitative thought of standard 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 worth difficulties, linear differential equations, Lyapunov balance, dynamical platforms and the Poincaré―Bendixson theorem, and bifurcation idea, and second-order themes together with oscillation concept, boundary worth difficulties, and Sturm―Liouville difficulties. The presentation is obvious and easy-to-understand, with figures and copious examples illustrating the that means of and motivation at the back of definitions, hypotheses, and normal theorems. A thoughtfully conceived collection of workouts including solutions and tricks make stronger the reader's realizing of the fabric. must haves are constrained to complex calculus and the user-friendly concept of differential equations and linear algebra, making the textual content compatible for senior undergraduates in addition.

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**Example text**

2) we have that for α < t ≤ aN |φ(t) − xα | ≤ |φ(t) − φ(aN )| + |φ(aN ) − xα | < . This implies that limt→α+ φ(t) = xα . 1, we see that φ(t) can be extended to t = α. 1 implies that if (α, φ(α)) or (β, φ(β)) is an accumulation point of the solution curve of (V) in D, then the solution can be extended further to a larger interval. 1. An interval J is called the maximal interval of existence of a solution φ(t) of IVP (V) if φ(t) exists on J and there does not exist an interval J1 J such that φ(t) has an extension from J to J1 .

1 t and x2 (t) = . 1. (i) Let x1 (t) = c1 x1 (t) + c2 x2 (t) = c1 + c2 t 0 = 0 0 for all t ∈ R implies that c1 = c2 = 0. The Wronskian of n functions of n components as deﬁned below is often used to discuss the linear dependence and linear independence of the functions. 2. Let xi = [xi1 , . . , xin ]T : Then the Wronskian of x1 , . . , xn is deﬁned by ⎡ x11 W (t) = W [x1 , . . , xn ](t) := det ⎣ · · · x1n (a, b) → Rn , i = 1, . . , n. ··· ··· ··· ⎤ xn1 · · · ⎦ (t). 1. Let xi : (a, b) → Rn , i = 1, .

Jl A with ⎡ λ1 ⎢ J0 = ⎣ ⎡ ⎤ .. ⎢ ⎢ and Ji = ⎢ ⎢ ⎣ ⎥ ⎦ . λq λq+i 1 .. ⎤ .. .. q×q ⎥ ⎥ ⎥ ⎥ 1 ⎦ λq+i ,i = 1, . . , l, ri ×ri satisfying q+r1 +· · ·+rl = n. We also know that all the λ’s in Ji , i = 0, . . , they are roots of the equation det(λI − A) = 0. 2, Part (d) and the results in the above special cases 1 and 2. 1) eAt = e(T JT −1 )t = T eJt T −1 ⎤ ⎡ Jt e 0 ⎢ ⎢ =T⎢ ⎣ ⎥ ⎥ −1 ⎥T , ⎦ e J1 t .. e Jl t where e J0 t ⎡ λt e 1 ⎢ =⎣ ⎤ .. ⎥ ⎦ . eλq t 46 2. LINEAR DIFFERENTIAL EQUATIONS ⎡ and e Ji t ⎢ ⎢ ⎢ ⎢ λq+i t ⎢ =e ⎢ ⎢ ⎢ ⎣ 1 t2 2!