By Stephen A. Wirkus, Visit Amazon's Randall J. Swift Page, search results, Learn about Author Central, Randall J. Swift,

"Featuring real-world functions from engineering and technology fields, A direction in traditional Differential Equations is the 1st booklet on usual differential equations (ODEs) to incorporate suitable laptop code and directions of MATLAB®, Mathematica®, and Maple. The booklet embeds the pc algebra code all through, providing the syntax subsequent to the proper concept. It absolutely describes approximations used to obtain�Read more...

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25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. y 2 +2xy x2 dy 2x2 dx = x2 + dy dx = y2 xy − y = x2 + y 2 (x + 2y)dx − xdy = 0 (y 2 − 2xy)dx + x2 dy = 0 2x3 y = y(2x2 − y 2 ) (x2 + y 2 )y = 2xy xy − y = x tan( xy ) (2x + y)dx − (4x + 2y)dy = 0 y 2 + x2 y = xyy x − y + (y − x)y = 0 (x + 4y)y = 2x + 3y (x − y)dx + (x + y)dy = 0 ydx = (2x + y)dy y y = 2( x+y )2 39. 2xdy + (x2 y 4 + 1)ydx = 0 40. ydx + x(2xy + 1)dy = 0 41. A function F is called homogeneous of degree n if F (tx, ty) = tn F (x, y) for all x and y.

4. EXACT EQUATIONS 34. 35. 36. 37. 38. 39. 4 43 miles during the ﬁrst hour after noon and traveled one mile during the second hour after noon. At what time did it begin snowing? A raft is being slowed down by resistance of the water, the resistance being proportional to the speed of the raft. 5 m/sec and at the end of 4 sec was 1 m/sec, when will the speed decrease to 1 cm/sec? What total distance will the raft travel? The amount of light absorbed by a thin layer of water is proportional to the amount of incident light and to the thickness of the layer.

5! For small x, the ﬁrst three terms of this expansion give a good approximation of sin x. 2 away from the known value. 31. Consider the Taylor expansion of ex about x = 0: sin x = x − x3 x2 + + ··· . 2! 3! 2 away from the known value. 32. Consider the function x f (x) = , x > 1, 1 − x − x2 and the approximations to it −1 1 −1 1 2 −1 1 2 3 f1 (x) = + 2 , f2 (x) = + 2 − 3 , f3 (x) = + 2 − 3 + 4. x x x x x x x x x Which of the functions is the better approximation for large x-values? Based on the pattern of the three approximation functions, can you write a function that has six terms and is a better approximation than each of the three functions given here?