By Qingkai Kong

ISBN-10: 3319112384

ISBN-13: 9783319112381

ISBN-10: 3319112392

ISBN-13: 9783319112398

This textual content is a rigorous therapy of the elemental qualitative thought of standard differential equations, at first graduate point. Designed as a versatile one-semester path yet providing adequate fabric for 2 semesters, a brief path covers middle issues equivalent to preliminary price difficulties, linear differential equations, Lyapunov balance, dynamical structures and the Poincaré—Bendixson theorem, and bifurcation conception, and second-order subject matters 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 in the back of definitions, hypotheses, and common theorems. A thoughtfully conceived choice of workouts including solutions and tricks make stronger the reader's realizing of the cloth. must haves are restricted to complicated calculus and the trouble-free idea of differential equations and linear algebra, making the textual content appropriate for senior undergraduates as well.

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**Example text**

N−1 dkj Ak . k=0 Note that the double sum above is norm-convergent for any t ∈ R. We can interchange the order of summation to obtain that ⎞ ⎛ n−1 ∞ n−1 j d t kj At k ⎠ ⎝ e = αk (t)Ak , A := j! j=0 k=0 k=0 where ∞ αk (t) = j=0 dkj tj , k = 0, . . , n − 1. j! Let J be the Jordan canonical form of matrix A. Then J and A have exactly the same eigenvalues. 4) e Jt αk (t)J k . = k=0 Let λi be an eigenvalue of A (and hence eigenvalue of J) with algebraic multiplicity mi . 4) that λi is a root of the equation eλt = r(λ, t) with algebraic multiplicity mi .

Clearly, ¯ k ⊂ Dk+1 for k ∈ N and ∪∞ Dk = D. D ∗ k=1 ∗ Let k ∈ N. It is easy to see that for any (t , x ) ∈ Dk , the set Gk := {(t, x) ∈ D : |t − t∗ | ≤ δk and |x − x∗ | ≤ δk } , where δk = √1 2 Mk = 1 k − 1 k+1 , is contained in Dk+1 . Let max ¯ k+1 (t,x)∈D {|f (t, x)|} and γk = min {δk , δk /Mk } . 2, the IVP x = f (t, x), x(t∗ ) = x∗ has a solution which exists for |t − t∗ | ≤ γk . , the mth extension extends the solution from [am−1 , bm−1 ] to [am , bm ] with am − am−1 ≤ −γk and/or bm − bm−1 ≥ γk , as long as the endpoint (am , φ(am )) and/or (bm , φ(bm )) is in Dk .

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 , . . , xm are linearly dependent on (a, b) means that one of x1 (t), . . , xm (t) can be expressed as a linear combination of the others with the same combination coeﬃcients for all t ∈ (a, b); (ii) x1 , . . 1) holds for all t ∈ (a, b), then c1 = · · · = cm = 0.

### A Short Course in Ordinary Differential Equations by Qingkai Kong

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