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Mechanical Engineers Handbook Episode 14

Tham khảo tài liệu 'mechanical engineers handbook episode 14', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 1. Differential Equations 769 EXAMPLE 1.33 Find the first three successive approximations y1 y2 y3 for the solution of the equation dyIdx 1 xy2 y 0 0 in a rectangle D 1 x 2 1 1 m 2 y 2 Solution We can write that f x y 1 xy2 maxD I f x y 1 1 9 that is 9 1 4 4 1 M f h min a b M min 2 9 9 @f @y 2xy maxD f @y 1 2 that is L 1. If we choose y0 x 0 then y1 X f x i .id v 0 X dx x 0 Differential Equations y2 x f x y1 dx 0 1 x3 dx x 0 4 y3 x1 f x y2 dx 42 x4 1 x x I 4 c 3 x 7 x10 dx x _ _ _. The absolute error in the third approximation does not exceed the value _ ML3 4 9 1 3 4 41 2.286 y y3 irh 8 V V 4 104 . EXAMPLE 1.34 Find the first three successive approximations for solution of the system y z y 0 0 z y2 z 0 1 1 x 1 D 1 y 1 1 z 1. m Solution The functions f x y z z and g x y z y2 are continuous in D. maxD If x y z maxD zI 1 maxD g x y z maxD I y21 1 that is M 1 f 0 max f @y @y 0 @z max @f @z 1 function f satisfies the Lipschitz condition with f 1 Lf 1 @ẵ @y @g @y 2y 2 2y @g -ị-2- 0 max dz @g @y 0 function g satisfies the Lipschitz condition with Lg 2. 770 Appendix Differential Equations and Systems of Differential Equations Then h min a Ệ1 vù 1 MM Differential Equations There is a unique solution y y x z z x x 2 1 1 of the initial system. Writing y0 x 0 z0 x 2 then yields yn x zn x x x y0 f t yn 1 t zn 1 t dt zn 1 t dt x1 z0 g t y. Ml- zn 1 t dt 2 x y2 1 t dt 0 y1 x y2 x y3 x fx . . . fx 1. x . . 1 fx 1 z0 t dt J 2 dt 2 z1 x 2 0dt 2 Zj t dt 1 dt x z2 x 1 if dt 1 x3 0 J0 2 2 2V 7 2 J0 4 2 12 x . rz 1 t3 . x x4 _ 1 rz t2 . 0 z2 t dt 0 12J dt 2 48 z3 x 2 ự 4 dt 1 x3 2 12 THEOREM 1.6 1.3.2 EXISTENCE AND UNIQUENESS THEOREM FOR DIFFERENTIAL EQUATIONS NOT SOLVED FOR DERIVATIVE In this section let us consider equations of the form F x y y0 0 1 126 It is obvious that generally for such equations not one but several integral curves pass through some point x0 y0 since as a rule when solving the equation F x y y0 0 for y0 we nd several not one real values y fị x y i 1 2 . and if .