Download e-book for iPad: A Short Course in Ordinary Differential Equations by Qingkai Kong

By Qingkai Kong

ISBN-10: 3319112392

ISBN-13: 9783319112398

This article is a rigorous therapy of the fundamental qualitative idea of normal differential equations, before everything graduate point. Designed as a versatile one-semester direction yet delivering sufficient fabric for 2 semesters, a quick direction covers middle themes equivalent to preliminary worth difficulties, linear differential equations, Lyapunov balance, dynamical structures and the Poincaré—Bendixson theorem, and bifurcation idea, and second-order issues together with oscillation concept, boundary price 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 at the back of definitions, hypotheses, and basic theorems. A thoughtfully conceived choice of workouts including solutions and tricks toughen the reader's knowing of the cloth. necessities are constrained to complex calculus and the undemanding concept of differential equations and linear algebra, making the textual content appropriate for senior undergraduates in addition.

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Extra info for A Short Course in Ordinary Differential Equations (Universitext)

Example text

Perhaps one of the oldest ways to obtain multiplicity results relies on the study of the linearization of the equation along a known solution. Such results, obtained by H. Amann [14, 15] and K. 3. In Section VIII-3, we study positive solutions of parametric problems u" + sf(t, u) = 0, u(0) = 0, u(l) = 0, where s > 0 and f(t, u) > 0 can be singular in t = 0 and 1. The results of this section can be found in M. Gaudenzi and P. Habets [125]. The problem u" + g{u)v! + f(t, u) - h(t), u(a) = u{b), u'(a) = u'{b), 9.

3) with u(0) < 0 are defined on a maximum interval of length i 2 r <2cosh~ 1 -=. V3 From a phase plane analysis, we can see that the orbit of u goes through some point (UQ,0). 3) such that v(0) = 2 - y/E~^T, v'(0) = 0, where E = £(u(0),u'(0)). Notice that (v(t) - 2)2 < E < 1 + (v{t) - 2)2 and v'(t) < 0 for t > 0. Hence, we compute for such t and t<- ,. - where w0 = ^ 2 )2- ( £ ? - ! ) = / = "<*> dr - 7 = 5 = = cosh WW r i«o ^ = 1 and w(t) = ^ ^ ^ < ^ ^ = < ^ . Hence, we have Repeating the computations in reverse time, the claim follows.

C. Lazer [198] who introduced the so-called Landesman-Lazer conditions. Here, a boundary value problem associated with the equation u" + f(t,u) = h(t), is said to be non-resonant for cases where it has a solution for every h and resonant if the problem is solvable only for some h. The use of lower and upper solutions can be useful to treat such situations when the nonlinearity interacts with the first eigenvalue. This matter is the subject of Chapter VII. L. W. Warner [177] in 1975 for the results on the left of the first eigenvalue.

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A Short Course in Ordinary Differential Equations (Universitext) by Qingkai Kong

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