By Peter V. O'Neil

ISBN-10: 0495082376

ISBN-13: 9780495082378

ISBN-10: 0935337865

ISBN-13: 9780935337860

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**Example text**

No w aN _ = 4y + 2 and ax aM aY = 3x, and 3x=4y+ 2 is satisfied by all (x, y) on a straight line . However, aN/ax = aM/ay cannot hold for all (x, y) i n an entire rectangle in the plane . Hence this differential equation is not exact on any rectangle . 1 9 Consider ex sin(y)-2x+(e x cos(y)+1)y'=0 . With M(x, y) = ex sin(y) - 2x and N(x, y) = e x cos(y) + 1, we hav e ON _ aM e` cos(y) ax = = ay for all (x, y) . Therefore this differential equation is exact . To find a potential function, set ** = ex sin(y) - 2x and ** = ex cos (y) + 1 .

Ex sin(y2) + xe x sin(y2) + (2xyex sin(y2) + eY)y ' = 7. sinh(x) sinh(y) + cosh(x) cosh(y)y' = 0 8. 4y4 + 3 cos (x) + (16y3x - 3 cos (y))y' = 0 9. 3y4 - 1 + 12xy3 Y' = 0; Y( l ) = 2 10. 2y - y2 sec2 (xy 2) + (2x - 2xysec 2 (xy 2))y' = 0; Y( l ) = 2 11. xcos(2y - x) - sin(2y - x) - 2xcos(2y - x)y'=0 ; y(7r/12) = *r/ 8 y(4) = 4 14. eY + (xeY -1)y' = 0 ; y(5) = 0 0 In each of Problems 9 through 14, determine if the differential equation is exact in some rectangle containing in it s interior the point where the initial condition is given .

Assume that a sphere of ice melts at a rate proportional to its surface area, retaining a spherical shape . Interpret melting as a reduction of volume wit h respect to time . Determine an expression for the volume of the ice at any time t . 20. A radioactive element has a half-life of ln(2) weeks . If e 3 tons are present at a given time, how much wil l be left 3 weeks later ? 21. 5 years . How much of a 10-kilogram bloc k of U-238 will be present 1 billion years from now ? 22. 1 grams in 4 minutes, what is the half-life of thi s element ?

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