tailieunhanh - Handbook of mathematics for engineers and scienteists part 201

Handbook of mathematics for engineers and scienteists part 201. 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. | 1368 Systems of Partial Differential Equations 7. d2u d2u dx2 dy2 uf u2 w2 - wg u2 w2 d2w d2w dx2 dy2 wf u2 w2 ug u2 w2 . 1 . A periodic solution in y u r x cos 0 x C1y C2 w r x sin 0 x C1y C2 where C1 and C2 are arbitrary constants and the functions r r x and 0 0 x are determined by the autonomous system of ordinary differential equations rxx r 0 x 2 C2r rf r2 r0Xx -2rX 0X rg r2 . 2 . Solution generalizes the solution of Item 1 u r z cos 0 z C1y C2 w r z sin 0 z C1y C2 z k1x k2y where C1 C2 k1 and k2 are arbitrary constants and the functions r r z and 0 0 z are determined by the autonomous system of ordinary differential equations k2 k2 yr z k2r 0Z 2 r k2 0 z C1 2 rf r2 k2 k2 r0 . z -2 k2 k2 0Z C1k2 r z rg r2 . 8. d2u d2u dx2 dy2 uf u2 - w2 wg u2 - w2 d2w dx2 d2w dy2 wf u2 - w2 ug u2 - w2 . Solution u r z cosh 0 z C1y C2 w r z sinh 0 z C1y C2 z k1x k2y where C1 C2 k1 and k2 are arbitrary constants and the functions r r z and 0 0 z are determined by the autonomous system of ordinary differential equations L2 1 L2t1 l22 12 1 i---. 12 ftT 2t k1 k2 rzz k1 r 0z i k20z C1 rf r -2 1 L2 V 7 ItL2 1 y2 1 It vnix 2 k1 k2 r0zz 2 L k1 k2 0z C1k2 z y . . Systems of the Form - - xn - F u w dt2 xn dx x dx F u w xn -Ï G u w dt2 xndx x dx G u w . Arbitrary functions depend on a linear combination of the unknowns. d2u ad du 1. Ï2 i xn J uf bu - cw g bu - cw d2w ad i dw q T xn wf bu - cw h bu - cw . 1 . Solution u t c0 x t w t b0 x t . Nonlinear Systems of Two Second-Order Equations 1369 where t and t are determined by the autonomous system of ordinary differential equations Vtt Vf bv - cf g b - cf tt ff bv - cf h bv - cf and the function 0 0 x t satisfies linear equation d2e dt2 a d i n de xn dx dx J f bp - cf e. For f const this equation can be solved by separation of variables. 2 . Let us multiply the first equation by b and the second one by -c and add the results together to obtain It a7r xn Zf Z bg Z - ch Z Z bu - cw. 1 dt2 xn dx dx J This equation will be