By Peter V O'Neil

ISBN-10: 1111427410

ISBN-13: 9781111427412

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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. 5 Additional Applications 31 Suppose that at time zero the object is dropped (not thrown) downward, so v(0) = 0.

4 Homogeneous, Bernoulli, and Riccati Equations 27 or x du = f (u) − u. dx The variables u and x separate as 1 1 du = d x. f (u) − u x We attempt to solve this separable equation and then substitute u = y/x to obtain the solution of the original homogeneous equation. 15 We will solve xy = y2 + y. x Write this as y = y x 2 y + . x With y = ux, this becomes xu + u = u 2 + u or xu = u 2 . The variables separate as 1 1 du = d x. 2 u x Integrate to obtain 1 − = ln |x| + c. u Then u= −1 . ln |x| + c y= −x , ln |x| + c Then and this is the general solution of the original homogeneous equation.

Cos(x + y) + sin(x − y)]y = cos(2x) In each of Problems 17 through 21, solve the initial value problem. 17. x y 2 y = y + 1; y(3e2 ) = 2 18. y = 3x 2 (y + 2); y(2) = 8 19. ln(y x )y = 3x 2 y; y(2) = e3 2 20. 2yy = e x−y ; y(4) = −2 21. yy = 2x sec(3y); y(2/3) = π/3 22. An object having a temperature of 90◦ Fahrenheit is placed in an environment kept at 60◦ . Ten minutes later the object has cooled to 88◦ . What will be the temperature of the object after it has been in this environment for 20 minutes?

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