By Peter V. O'Neil
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Extra info for Advanced Engineering Mathematics, 6th Edition
Ln yx y = 3x2 y y 2 = e3 2. y + xy = 0 14. 2yy = ex−y y 4 = −2 3. cos y y = sin x + y 15. yy = 2x sec 3y y 2/3 = /3 4. e x+y y = 3x 5. xy + y = y 6. y = 2 x + 1 2 − 2y 2y 7. x sin y y = cos y 8. 2y2 + 1 x y = y x+1 9. y + y = ex − sin y 10. cos x + y + sin x − y y = cos 2x 2 16. An object having a temperature of 90 degrees Fahrenheit is placed into an environment kept at 60 degrees. Ten minutes later the object has cooled to 88 degrees. What will be the temperature of the object after it has been in this environment for 20 minutes?
20 The equation y2 − 6xy + 3xy − 6x2 y = 0 is not exact on any rectangle. 12) have the same solution. 12) is just y y2 − 6xy + 3xy − 6x2 y = 0 and if y = 0, then necessarily y2 − 6xy + 3xy − 6x2 y = 0. 11) as well. To review what has just occurred, we began with a nonexact differential equation. We multiplied it by a function chosen so that the new equation was exact. We solved this exact equation, then found that this solution also worked for the original, nonexact equation. The function therefore enabled us to solve a nonexact equation by solving an exact one.
In this case, we can solve for y x explicitly. Begin by taking the exponential of both sides to obtain 1 + y = ek e−1/x = Ae−1/x in which we have written A = ek . Since k could be any number, A can be any positive number. Then 1 + y = ±Ae−1/x = Be−1/x in which B = ±A can be any nonzero number. The general solution is y = −1 + Be−1/x in which B is any nonzero number. Now revisit the assumption that x = 0 and y = −1. In the general solution, we actually obtain y = −1 if we allow B = 0. Further, the constant function y x = −1 does satisfy x2 y = 1 + y.
Advanced Engineering Mathematics, 6th Edition by Peter V. O'Neil