By Samuel Zaidman

ISBN-10: 0273084275

ISBN-13: 9780273084273

ISBN-10: 0822484277

ISBN-13: 9780822484271

ISBN-10: 1584880112

ISBN-13: 9781584880110

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Extra resources for Abstract differential equations

Example text

3. (i) ln x/(x−1). (ii) (1/2)(1−x) ln[(x−2)/x]−1. (iii) e−x . (iv) (x/2)× ln[(1 + x)/(1 − x)] − 1. 4. (i) Verify directly. 5). 5. e−δx /2 x δt2 /2 e dt. 6. Use y2 (x) = y1 (x)y(x) and the fact that y1 (x) and y2 (x) are solutions. 7. (i) Use Abel’s identity. (ii) If both attain maxima or minima at x0 , then φ1 (x0 ) = φ2 (x0 ) = 0. (iii) Use Abel’s identity. (iv) If x0 is a common point of inﬂexion, then φ1 (x0 ) = φ2 (x0 ) = 0. (v) W (x∗ ) = 0 implies φ2 (x) = cφ1 (x). If φ1 (x∗ ) = 0, then φ1 (x) ≡ 0, and if φ1 (x∗ ) = 0 then c = φ2 (x∗ )/φ1 (x∗ ).

7. Suppose that y = y(x) is a solution of the initial value problem y = yg(x, y), y(0) = 1 on the interval [0, β], where g(x, y) is a bounded and continuous function in the (x, y) plane. Show that there exists a constant C such that |y(x)| ≤ eCx for all x ∈ [0, β]. 8. Suppose α > 0, γ > 0, c0 , c1 , c2 are nonnegative constants and u(x) is a nonnegative bounded continuous solution of either the inequality x u(x) ≤ c0 e−αx + c1 0 ∞ e−α(x−t) u(t)dt + c2 e−γt u(x + t)dt, 0 x ≥ 0, or the inequality 0 u(x) ≤ c0 eαx + c1 0 eα(x−t) u(t)dt + c2 If β = eγt u(x + t)dt, −∞ x x ≤ 0.

Note that β(β + 2r) → 0 as β → 0. 15. c1 x + (c2 /x) + (1/2)[(x − (1/x)) ln(1 + x) − x ln x − 1]. Lecture 7 Preliminaries to Existence and Uniqueness of Solutions So far, mostly we have engaged ourselves in solving DEs, tacitly assuming that there always exists a solution. However, the theory of existence and uniqueness of solutions of the initial value problems is quite complex. 1) where f (x, y) will be assumed to be continuous in a domain D containing the point (x0 , y0 ). 1) in an interval J containing x0 , we mean a function y(x) satisfying (i) y(x0 ) = y0 , (ii) y (x) exists for all x ∈ J, (iii) for all x ∈ J the points (x, y(x)) ∈ D, and (iv) y (x) = f (x, y(x)) for all x ∈ J.