tailieunhanh - Handbook of mathematics for engineers and scienteists part 183

Handbook of mathematics for engineers and scienteists part 183. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1242 Systems of Ordinary Differential Equations 5. x lt a i x x 3f y x y a t y y 3g y x . Generalized Ermakov system. 1 . First integral 2 xy t - yx t 2 i uf u - u 3g u du C where C is an arbitrary constant. 2 . Suppose p Q t is a nontrivial solution of the second-order linear differential equation a t V 0. 1 Then the transformation dt x y T J 2 t U t V Q t leads to the autonomous system of equations u T u 3f v u v T v- g v u . 3 3 . Particular solution of system 3 is u As C2 t 2 C1T Co v Aks C2T2 Cit Co f k 11 4 C0C2 - 4C12_ A where C0 C1 and C2 are arbitrary constants and k is a root of the algebraic transcendental equation k4f k g k . 6. xtt f yt xt ytt g yt xt . 1 . The transformation u x t w yt 1 leads to a system of the first-order equations u t f w u wt g w u . 2 Eliminating t yields a homogeneous first-order equation whose solution is given by w u f 9 î - f ln u C 3 where C is an arbitrary constant. On solving 3 for w one obtains w w u C . On substituting this expression into the first equation of 2 one can find u u t and then w w t . Finally one can determine x x t and y y t from 1 by simple integration. . Nonlinear Systems of Two Equations 1243 2 . The Suslov problem. The problem of a point particle sliding down an inclined rough plane is described by the equations Ed - x h 1 t ytt - f y x t 2 y t 2 yj xt 2 y t 2 which correspond to a special case of the system in question with f z 1 - 9 Z - . v 1 z2 v 1 z2 The solution of the corresponding Cauchy problem under the initial conditions x 0 y 0 x t 0 0 yt 0 1 leads for the case k 1 to the following dependences x t and y t written in parametric form - -L _Ly4 y 2 ly ly3 t-1 ly2 Im y 0 y 1 x - 16 16y - 4 ln y y 3 - 2 y - 6 y t 4 - 4 y - 2 ln y 0 y 1 . 7. Xtt x x y t xt yt y t y x y t x t y t . 1 . First integral xyt- yxt C where C is an arbitrary constant. Remark. The function can also be dependent on the second and higher derivatives with respect to t. 2 . Particular solution y C1 x where C1 is an .