tailieunhanh - Handbook of mathematics for engineers and scienteists part 197

Handbook of mathematics for engineers and scienteists part 197. 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. | 1340 Systems of Partial Differential Equations 2 . If k 1 it is more beneficial to seek solutions in the form u x k-1 p Z w x Z t a In x where a is an arbitrary constant and the functions q Z and are determined by the autonomous system of ordinary differential equations art -1- p k f m Z 1 - k m . 12. uf unwm wg unwm . Solution U y w e-ra kx- t z a _ where k a 3 and A are arbitrary constants and the functions y and z are determined by the autonomous system of ordinary differential equations ayZ kmy yf ynzm -ßz nXz zg yn zm . 13. uf unwm W wg ukws . ox ot Let A sn - km 0. Multiplicative separable solutions u v x s A A A w v x -k A t n A where the functions v x and t are determined by the autonomous ordinary differential equations - vX vf v - g rt Integrating yields s f dv _ n d _ a J vf v 1 a g 2 14. ---- au In u uf unwm ----- wg unwm . dx dt Solution u exp Cme x y w exp -Cne x z kx - At where C k and A are arbitrary constants and the functions y and z are determined by the autonomous system of ordinary differential equations kyZ ay ln y yf ynzm -AzZ zg yn zm . . Linear Systems of Two Second-Order Equations 1341 15. uf auk bw uk. dx V dt Solution w p x C exp -At k f bp x dx dw 1 k dt J where p x is an arbitrary function and C is an arbitrary constant. 16. ---- uf aun bw dx Solution dw dt ukg aun bw . i U C11 C2 n-k 6 x w x - a Cit C2 n k 0 x n where Ci and C2 are arbitrary constants and the functions 0 0 x and p x are determined by the system of differential-algebraic equations ox 0f bp nn-k b k n 1 on k g bv . aC1n du cwnu dx a bwn General solution with b 0 17. aw bwn 1 uk. dt fi t eF x - -i-1 n eF x x dx w Wt u A b a U ---------1----T aw bwn 1 where x and t are arbitrary functions. ck f w x - a p x dx . Linear Systems of Two Second-Order Equations 1. L X f b u cw W b2u c2w. dt dx2 dt dx2 Constant-coefficient second-order linear system of parabolic type. Solution u b1 A2 11 b1 A1 eX21A b2 A1 A2 1 b2 A1 A2 2 w 1 x eX1t01 eX2t02 A1 A2 where A1 and