Tham khảo tài liệu 'handbook of industrial automation - richard l. shell and ernest l. hall part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | and becomes - Ề - -l dx or dv f- V 1 dx This is a first-order linear equation for which we have a general formula for the solution. We have here P x 1 and Q x 1. Hence y 1 V e ex dx c - V 1 1 1 Ce x y V 1 y 1 Ce-X ex c Cf-xfi- 12 1 ax and y 2 X c2 If p 1 - V .VC - 1 dx and if X 0 y Cje- 2 2 X c2 If X 0 y -Ci e x2 2 X c2 Independent Variable Absent The procedure is to take y to be the independent variable and let y be the new dependent variable. Dependent Variable Absent Example 8 xy 2 _ k 1 0 The dependent variable y is absent. Let p y then p y and the equation becomes xp x2 - l p - 1 0 p X 0 p -1 dp A X---I dx 0 X p X - Ỉ 0 or p 1 X In p 11 In x c m ị c I x 2 f c x fi 2 where Cj 0 x p - 1 C xc If p 1. ặ Cj x e p 2 1 dx If X 0 y y X c2 If X 0 Example 9. yy y Ỷ 1 0. Let y p. Then dp _dp dy _ dp dx dy dx dy The equation becomes dp 2 yp i . p1 1 0 dy dp 1 p 0 dy p dp dy 0 p Vp y pdp dy_ f . p 1 y lln p2 1 In 1 1 c ln 2 1 In y2 c ln 2 l 2 c 1 fli y r2 - ị - - y where K ec 0 y2 y2 p yK - y2 dx p y dx - K - y2 1 2 X c Copyright 2000 Marcel Dekker Inc. K y2 x C 2 I 2 K x C 2 In the preceding pages we have given a sampling of the many ingenious techniques which have been developed to study particular classes of first-order differential equations. More complete discussions including extensions of the techniques we have described and other techniques can be found in standard textbooks. There is an excellent discussion in Ref. 5 Chap. 1. Very extensive and thorough treatments are given in Refs 68. Each of these references presents a large number of ordinary differential equations and then solutions. Reference 8 is a more extensive compilation than Refs 6 or 7 but Refs 6 and 7 contain more theory and references. Figure 1 A GEOMETRICAL APPROACH We have been describing methods for obtaining explicit formulas for solutions of differential equations. There are however other ways of obtaining useful information about solutions. A geometrical approach yields a .