By Ravi P. Agarwal

ISBN-10: 0387712755

ISBN-13: 9780387712758

ISBN-10: 0387712763

ISBN-13: 9780387712765

This textbook offers a rigorous and lucid creation to the speculation of normal differential equations (ODEs), which function mathematical types for lots of fascinating real-world difficulties in technological know-how, engineering, and different disciplines.

Key good points of this textbook:

Effectively organizes the topic into simply plausible sections within the kind of forty two class-tested lectures

Provides a theoretical therapy by way of organizing the cloth round theorems and proofs

Uses designated examples to force the presentation

Includes a number of workout units that inspire pursuing extensions of the cloth, each one with an "answers or tricks" section

Covers an array of complicated subject matters which enable for flexibility in constructing the topic past the basics

Provides first-class grounding and thought for destiny learn contributions to the sector of ODEs and similar areas

This ebook is perfect for a senior undergraduate or a graduate-level path on usual differential equations. must haves contain a direction in calculus.

Series: Universitext

Ravi P. Agarwal acquired his Ph.D. in arithmetic from the Indian Institute of know-how, Madras, India. he's a professor of arithmetic on the Florida Institute of expertise. His study pursuits contain numerical research, inequalities, mounted element theorems, and differential and distinction equations. he's the author/co-author of over 800 magazine articles and greater than 20 books, and actively contributes to over forty journals and ebook sequence in quite a few capacities.

Donal O’Regan acquired his Ph.D. in arithmetic from Oregon nation college, Oregon, U.S.A. he's a professor of arithmetic on the nationwide college of eire, Galway. he's the author/co-author of 14 books and has released over 650 papers on mounted element conception, operator, critical, differential and distinction equations. He serves at the editorial board of many mathematical journals.

Previously, the authors have co-authored/co-edited the next books with Springer: *Infinite period difficulties for Differential, distinction and indispensable Equations; Singular Differential and indispensable Equations with purposes; Nonlinear research and functions: To V. Lakshmikanthan on his 80 ^{th} Birthday. *In addition, they've got collaborated with others at the following titles:

*Positive strategies of Differential, distinction and critical Equations; Oscillation idea for distinction and practical Differential Equations; Oscillation idea for moment Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations.*

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**Extra resources for An Introduction to Ordinary Differential Equations**

**Sample text**

When each member of the second family cuts each member of the ﬁrst family according to a deﬁnite law, any curve of either of the families is said to be a trajectory of the family. The most important case is that in which curves of the families intersect at a constant angle. The orthogonal trajectories of a given family of curves are the curves that cut the given family at right angles. The slopes y1 and y2 of the tangents to the curves of the family and to the sought for orthogonal trajectories must at each point satisfy the orthogonality condition y1 y2 = −1.

7. 19) 42 Lecture 6 in J. Show the following: (i) If y1 (x) and y2 (x) vanish at the same point in J, then y1 (x) is a constant multiple of y2 (x). (ii) If y1 (x) and y2 (x) have maxima or minima at the same point in the open interval J, then y1 (x) and y2 (x) are not the linearly independent solutions. (iii) If W (y1 , y2 )(x) is independent of x, then p1 (x) = 0 for all x ∈ J. (iv) If y1 (x) and y2 (x) are linearly independent, then y1 (x) and y2 (x) cannot have a common point of inﬂexion in J unless p1 (x) and p2 (x) vanish simultaneously there.

Then the equation f (x, y) = 0 has a unique continuous solution y(x) in [α, β]. 1. 9) 50 Lecture 7 where f (x, y) is continuous in a domain D containing the point (x0 , y0 ), is equivalent to the integral equation x y(x) = y0 + (x − x0 )y1 + (x − t)f (t, y(t))dt. 2. 3), also ﬁnd the x y . (ii) . (iii) (i) (1 + x2 ) (1 + y 2 ) (iv) |xy|. (v) y + [x]. (vi) the following functions satisfy the Lipschitz constants: x2 cos2 y + y sin2 x. x2 y 2 + xy + 1. 3. By computing appropriate Lipschitz constants, show that the following functions satisfy the Lipschitz condition in the given domains: (i) x sin y + y cos x, |x| ≤ a, |y| ≤ b.

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