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