A Short Course in Ordinary Differential Equations by Qingkai Kong

A Short Course in Ordinary Differential Equations by Qingkai Kong

By Qingkai Kong

This article is a rigorous remedy of the elemental qualitative concept of standard differential equations, at the start graduate point. Designed as a versatile one-semester path yet providing sufficient fabric for 2 semesters, a quick direction covers center themes reminiscent of 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 conception, 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 in the back of definitions, hypotheses, and common theorems. A thoughtfully conceived choice of workouts including solutions and tricks make stronger the reader's knowing of the fabric. necessities are constrained to complex calculus and the undemanding idea of differential equations and linear algebra, making the textual content compatible for senior undergraduates in addition.

Show description

Read Online or Download A Short Course in Ordinary Differential Equations (Universitext) PDF

Best differential equations books

Boundary Value Problems: And Partial Differential Equations

Boundary price difficulties is the best textual content on boundary worth difficulties and Fourier sequence for execs and scholars in engineering, technological know-how, and arithmetic who paintings with partial differential equations. during this up-to-date variation, writer David Powers presents a radical evaluate of fixing boundary price difficulties regarding partial differential equations via the tools of separation of variables.

Invertible Point Transformations and Nonlinear Differential Equations

The invertible element transformation is a strong device within the learn of nonlinear differential and distinction questions. This ebook offers a entire creation to this system. usual and partial differential equations are studied with this strategy. The ebook additionally covers nonlinear distinction equations.

Dynamical systems and numerical analysis

This ebook unites the examine of dynamical platforms and numerical answer of differential equations. the 1st 3 chapters comprise the weather of the speculation of dynamical platforms and the numerical resolution of initial-value difficulties. within the closing chapters, numerical tools are formulted as dynamical structures and the convergence and balance houses of the tools are tested.

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

Example text

Proof. Let t ∈ (a, b). Since X (t) = lim τ →t X(τ ) − X(t) , τ −t for any τ ∈ (a, b) we have X(τ ) = X(t) + X (t)(τ − t) + o(τ − t), where o(τ − t)/(τ − t) → 0 as τ → t. Note that X(t) satisfies Eq. (H). 7) X(τ ) = X(t) + A(t)X(t)(τ − t) + o(τ − t) = [I + A(t)(τ − t)]X(t) + o(τ − t). 3. NONHOMOGENEOUS LINEAR EQUATIONS 37 By the definition of determinant we have det[I + A(t)(τ − t)] 1 + a11 (t)(τ − t) a12 (t)(τ − t) 1 + a22 (t)(τ − t) a21 (t)(τ − t) = ··· ··· an1 (t)(τ − t) an2 (t)(τ − t) ··· ··· ··· ··· a1n (t)(τ − t) a2n (t)(τ − t) ··· 1 + ann (t)(τ − t) n aii (t)(τ − t) + o(τ − t) = 1 + tr A(t)(τ − t) + o(τ − t).

Tn−1 (n−1)! . t2 2! t 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥. 4. 2, Part (b), eAt = eaIt eBt ⎡ 1 ⎢ ⎢ ⎢ ⎢ at ⎢ = e ⎢ ⎢ ⎢ ⎣ = eat I + Bt + t2 2! t .. .. .. ··· .. .. .. B n−1 tn−1 B 2 t2 + ···+ 2! (n − 1)! ⎤ n−1 t (n−1)! ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦ .. t2 2! t 1 For a general matrix A ∈ Rn×n , from linear algebra, see [27] for the n×n such that detail, we know that there ⎤ matrix T ∈ C ⎡ is a nonsingular J0 ⎥ ⎢ J1 ⎥ ⎢ T −1 AT = J, where J= ⎢ ⎥ is the Jordan canonical form of .. ⎦ ⎣ . Jl A with ⎡ λ1 ⎢ J0 = ⎣ ⎡ ⎤ ..

2 in [4], where the proof was not given. 2. Let λi , i = 1, . . , h, be the eigenvalues of a matrix A ∈ Rn×n whose algebraic multiplicities are mi , i = 1, . . , h, respectively. Then n−1 eAt = αk (t)Ak , k=0 where αk (t), k = 0, . . 3) ∂r ∂ mi −1 r (λi , t), . . , tmi −1 eλi t = (λi , t), i=1, . . , h, eλi t =r(λi , t), teλi t = ∂λ ∂λmi −1 with n−1 αk (t)λk . 2, we need the Cayley–Hamilton theorem below from linear algebra. 3 (Cayley–Hamilton Theorem). Let the characteristic polynomial of matrix A ∈ Rn×n be n−1 p(λ) = det(λI − A) = λn − ci λi , i=1 where ci ∈ R, i = 1, .

Download PDF sample

Rated 4.61 of 5 – based on 19 votes
Comments are closed.