tailieunhanh - Handbook of mathematics for engineers and scienteists part 199

Handbook of mathematics for engineers and scienteists part 199. 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. | 1354 Systems of Partial Differential Equations 2 . A periodic solution in time u r x cos 0 x C1t C2 w r x sin 0 x C1t 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 arfx - ar 0 x 2 rf r2 0 ar0fx 2ar x0x - C1r rg r2 0. 3 . Solution generalizes the solution of Item 2 u r z cos 0 z C1t C2 w r z sin 0 z C1t C2 z x At where C1 C2 and A are arbitrary constants and the functions r r z and 0 0 z are determined by the system of ordinary differential equations ar Zz - ar 0 z 2 - Ar z rf r2 0 ar0fz 2ar z0z - Ar0 z - C1r rg r2 0. a - uf u2 w2 - wg u2 w2 - w arctan dx2 u a-- wf u2 w2 ug u2 w2 u arctanf dx2 u du 20. ---- dt dw dt Functional separable solution for fixed t it defines a structure periodic in x u r t cos t x f t w r t sin t x f t where the functions r r t Q t and f t are determined by the autonomous system of ordinary differential equations r t -arQ2 rf r2 p t h r2 t h r2 g r2 . . Arbitrary functions depend on the difference of squares of the unknowns. du d 2u uf u2 - - w2 wg u2 - w2 21. a dx2 dw d 2w ug u2 - w2 wf u2 - w2 . a dt dx2 1 . Solution u t cosh Q x t w t sinh Q x t Q x t C1x J g f2 dt C2 where C1 and C2 are arbitrary constants and the function f t is described by the separable first-order ordinary differential equation t f 2 aC2 . Nonlinear Systems of Two Second-Order Equations 1355 whose general solution can be represented in implicit form as J ff 02 aC 3 2 . Solution u r x cosh 0 x Ci t C2 w r x sinh 0 x Ci t 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 ar Xcx ar 0 x 2 rf r2 0 ar0 x 2ar x 0x rg r2 - C1r 0. 3 . Solution generalizes the solution of Item 2 u r z cosh 0 z C1t C2 w r z sinh 0 z C11 C2 z x At where C1 C2 and A are arbitrary constants and the functions r r z and 0 0 z are determined by the autonomous system of ordinary .