By Qingkai Kong
This textual content 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 providing sufficient fabric for 2 semesters, a quick direction covers center issues equivalent to preliminary price difficulties, linear differential equations, Lyapunov balance, dynamical platforms and the Poincaré—Bendixson theorem, and bifurcation conception, and second-order issues together with oscillation conception, boundary worth difficulties, and Sturm—Liouville difficulties. The presentation is obvious and easy-to-understand, with figures and copious examples illustrating the that means of and motivation in the back of definitions, hypotheses, and common theorems. A thoughtfully conceived collection of routines including solutions and tricks make stronger the reader's figuring out of the cloth. necessities are constrained to complicated calculus and the ordinary concept of differential equations and linear algebra, making the textual content compatible for senior undergraduates as well.
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Additional info for A Short Course in Ordinary Differential Equations
Xn are linearly independent on (a, b). 3. 1 implies that the Wronskian W (t) of n linearly independent solutions of Eq. (H) satisﬁes W (t) ≡ 0 on (a, b) W (t) = 0 on (a, b) ⇐⇒ W (t0 ) = 0 for some t0 ∈ (a, b), and ⇐⇒ W (t0 ) = 0 for some t0 ∈ (a, b). Therefore, the linear dependence or independence of the solutions can be determined by the value of W (t) at one point t0 ∈ (a, b). 2. GENERAL THEORY FOR HOMOGENEOUS LINEAR EQUATIONS 35 (ii) Equation (H) has at least n linearly independent solutions.
X(t) = eA(t−t0 ) is the principal matrix solution of Eq. (H-c) at t0 . Proof. We ﬁrst show that the X(t) given in the theorem is a matrix solution of Eq. (H-c) on R. In fact, by deﬁnition, ∞ X(t) = k=0 Ak (t − t0 )k , k! t ∈ R. 4. HOMOGENEOUS LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS 43 For any r > 0 and t ∈ [t0 − r, t0 + r], (|A|r)k Ak (t − t0 )k ≤ . k! k! ∞ Since the scalar series k=0 |Ar|k /k! is convergent, the matrix-valued funck k tion series ∞ k=0 A (t − t0 ) /k! is uniformly convergent on [t0 − r, t0 + r].
2. 3 we see that for any A ∈ n−1 i Rn×n , there are ci ∈ R, i = 1, . . , n − 1, such that An = i=1 ci A . It follows that for any j ∈ N0 , Aj can be expressed as a linear combination of k I, A, . . , An−1 . Let Aj = n−1 k=0 dkj A for dkj ∈ R. Then ∞ eAt = j=0 Aj tj = j! ∞ j=0 tj j! 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, .
A Short Course in Ordinary Differential Equations by Qingkai Kong
Categories: Differential Equations