# Differential Equations with Boundary-Value Problems, 8th Edition

## Dennis G. Zill

Language: English

Pages: 664

ISBN: 1111827060

Format: PDF / Kindle (mobi) / ePub

** NOTE: This book DOES NOT come with Access Code **

DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 8th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This proven and accessible book speaks to beginning engineering and math students through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, definitions, and group projects. Written in a straightforward, readable, and helpful style, the book provides a thorough treatment of boundary-value problems and partial differential equations.

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copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 136 ● CHAPTER 4 HIGHER-ORDER DIFFERENTIAL EQUATIONS EXAMPLE 4 Solve Fourth-Order DE d 4y d 2y ϩ 2

as you passed the natural length of the cable. This could lead to discomfort, injury, or even a Darwin award. You want to choose the cord with a k value large enough to stop you above or just touching the water, but not too suddenly. Consequently, you are interested in finding the distance you fall below the natural length of the cord as a function of the spring constant. To do that, you must solve the differential equation that we have derived in words above: The force mxЉ on your body is given

EQUATIONS a natural inclination for most students (and instructors) to relax and be content. However, a solution of an initial-value problem might not be unique. We saw in Example 4 of Section 1.2 that the initial-value problem dy ϭ xy1/2, dx y(0) ϭ 0 (9) has at least two solutions, y ϭ 0 and y ϭ 161 x4. We are now in a position to solve the equation. Separating variables and integrating y Ϫ1/2 dy ϭ x dx gives y (0, 0) a=0 2y1/2 ϭ a>0 x x2 ϩ c1 2 x4 ϩ c , 2 2 c Ն 0. When x ϭ 0,

Example 5 we solve equation (8) of Section 1.3. EXAMPLE 5 Mixture of Two Salt Solutions Recall that the large tank considered in Section 1.3 held 300 gallons of a brine solution. Salt was entering and leaving the tank; a brine solution was being pumped into the tank at the rate of 3 gal/min; it mixed with the solution there, and then the mixture was pumped out at the rate of 3 gal/min. The concentration of the salt in the inflow, or solution entering, was 2 lb/gal, so salt was entering the

Ϫ x2. 25 25 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. ) 108 ● CHAPTER 3