By Qingkai Kong
This article is a rigorous remedy of the elemental qualitative idea of standard differential equations, first and foremost graduate point. Designed as a versatile one-semester path yet delivering sufficient fabric for 2 semesters, a quick path covers center subject matters corresponding to preliminary price difficulties, linear differential equations, Lyapunov balance, dynamical structures and the Poincaré—Bendixson theorem, and bifurcation concept, and second-order themes together with oscillation idea, boundary price difficulties, and Sturm—Liouville difficulties. The presentation is apparent and easy-to-understand, with figures and copious examples illustrating the which means of and motivation at the back of definitions, hypotheses, and basic theorems. A thoughtfully conceived collection of workouts including solutions and tricks strengthen the reader's figuring out of the cloth. must haves are constrained to complicated calculus and the undemanding thought of differential equations and linear algebra, making the textual content appropriate for senior undergraduates to boot.
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Additional resources for A Short Course in Ordinary Differential Equations (Universitext)
Clearly, the corresponding homogeneous linear equation is (h-n) x(n) + a1 (t)x(n−1) + · · · + an−1 (t)x + an (t)x = 0. 2. Let φ1 (t), . . , φn (t) be linearly independent solutions of Eq. (h-n) and ⎡ ⎤ φ1 φ2 ··· φn ⎢ φ1 φ2 ··· φn ⎥ ⎥ W [φ1 , . . , φn ](t) = det ⎢ ⎣ ··· ··· ··· · · · ⎦ (t). 4. HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 41 Then the general solution of Eq. (nh-n) is n n ci φi (t) + x(t) = i=1 t φk (t) k=1 f (s) t0 Wk [φ1 , . . , φn ](s) ds, W [φ1 , . . , φn ](s) where for k = 1, .
1. Let φ1 (t) and φ2 (t) be two linearly independent solutions of Eq. (h-2). Then the general solution of Eq. (nh-2) is t x = c1 φ1 (t) + c2 φ2 (t) + t0 where W (s) = det φ1 φ1 f (s) − φ1 (t)φ2 (s) + φ2 (t)φ1 (s) ds, W (s) φ2 (s). φ2 Proof. Let x1 = x and x2 = x . Then Eq. (nh-2) can be written as the vector-valued ﬁrst-order linear equation (NH-2) x1 x2 = 0 1 −a2 (t) −a1 (t) x1 0 . + x2 f (t) Clearly, the corresponding homogeneous linear equation to Eq. (NH-2) is (H-2) x1 x2 = 0 −a2 (t) 1 −a1 (t) x1 .
1 shows that the solutions of Eq. (H) form a vector space. To determine the dimension of this space, we need the following deﬁnitions and lemmas on linear dependence and independence. 1. Let xi = [xi1 , · · · , xin ]T : (a, b) → Rn , i = 1, . . , m. Then x1 , . . , xm are said to be linearly dependent on (a, b) if there exist c1 , . . 1) c1 x1 (t) + c2 x2 (t) + · · · + cm xm (t) ≡ 0 on (a, b). Otherwise, x1 , . . , xm are said to be linearly independent on (a, b). 2. 1. It is easy to see that (i) x1 , .
A Short Course in Ordinary Differential Equations (Universitext) by Qingkai Kong