Get An Introduction to Ordinary Differential Equations PDF

By Ravi P. Agarwal, Donal O'Regan

ISBN-10: 0387712763

ISBN-13: 9780387712765

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

Key good points of this textbook:

* successfully organizes the topic into simply plausible sections within the kind of forty two class-tested lectures
* presents a theoretical remedy via organizing the cloth round theorems and proofs
* makes use of particular examples to force the presentation
* comprises a variety of workout units that inspire pursuing extensions of the fabric, each one with an "answers or hints" section
* Covers an array of complicated issues which enable for flexibility in constructing the topic past the basics
* offers very good grounding and idea for destiny examine contributions to the sector of ODEs and comparable areas

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

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Extra info for An Introduction to Ordinary Differential Equations (Universitext)

Example text

For the family of parabolas y = ax2 , we have y = 2ax or y = 2y/x. Thus, the DE of the desired orthogonal trajectories is y = −(x/2y). Separating the variables, we find 2yy + x = 0, and on integrating this DE we obtain the family of ellipses x2 + 2y 2 = c. 1. 1) is both homogeneous and exact, and M x + N y is not a constant, then its general solution is given by M x + N y = c. 2. 1) is homogeneous and M and N possess continuous partial derivatives in some domain D, then M xMx + yMy . 3. Solve the following DEs: (i) x sin y + (x2 + 1) cos yy = 0.

7. Show that the introduction of polar coordinates x = r cos θ, y = r sin θ leads to separation of variables in a homogeneous DE y = f (y/x). Hence, solve the DE ax + by . 8. Solve y = y − xy 2 x + x2 y by making the substitution y = vxn for an appropriate n. 9. Show that the families of parabolas y 2 = 2cx + c2 , x2 = 4a(y + a) are self-orthogonal. 10. Show that the circles x2 +y 2 = px intersect the circles x2 +y 2 = qy at right angles. 11. 9) is separable. The requirement f (x, y) = 0 is essential, for this consider the function f (x, y) = x2 e2y , y ≤ 0 x2 ey , y > 0.

1), we shall also need a Gronwall’s-type integral inequality, which is contained in the following result. 3. Let u(x), p(x) and q(x) be nonnegative continuous functions in the interval |x − x0 | ≤ a and x u(x) ≤ p(x) + q(t)u(t)dt for |x − x0 | ≤ a. 5) x0 Then the following inequality holds: x u(x) ≤ p(x)+ x p(t)q(t) exp x0 q(s)ds dt for |x−x0 | ≤ a. 6) t Proof. 6) for x0 ≤ x ≤ x0 + a whereas for x0 − a ≤ x ≤ x0 the proof is similar. We define x r(x) = q(t)u(t)dt x0 so that r(x0 ) = 0, and r (x) = q(x)u(x).

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An Introduction to Ordinary Differential Equations (Universitext) by Ravi P. Agarwal, Donal O'Regan

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