Handbook of mathematics for engineers and scienteists part 200. 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. | . Nonlinear Systems of Two Second-Order Equations 1361 The first equation in 1 is a separable equation its solution can be written out in implicit form. The second equation in 1 can be solved using the change of variable ez it is reduced to a linear equation for Z . Equation 2 admits exact solutions of the form 0 exp a2 t x2 a0 t where the functions an t are described by the equations a 2 f ct2 4aa2 a o f o 2a n 1 2. This system can be successively integrated since the first equation is a Bernoulli equation and the second one is linear in the unknown. If f const equation 2 also has a traveling-wave solution 0 0 kx - At . . Arbitrary functions depend on the product of powers of the unknowns. 8. du dt a d xn dx i n du I xn- dx uf x ukwm dw dt b d xn dx xn dw dx wg x ukwm . Multiplicative separable solution u e mXty x w ekXtz x where A is an arbitrary constant and the functions y y x and z z x are determined by the system of ordinary differential equations ax-n xn y x X mAy yf x yk zm 0 bx-1 xnz x X - kAz zg x ykzm 0. 9. du a 9 1 n du xn dx u1 knf unwm dt xn dx dw _ b d n dwy 1 xn dx . w1-kmg unwm dt xn dx Self-similar solution u Cit C2 kn y w C11 C2 km z x VC it C2 where C1 and C2 are arbitrary constants and the functions y y and z z are determined by the system of ordinary differential equations arxr y é 1 Ci C y y1 knf ynzm .zX X 1 CitzX - kmz z1 . 1362 Systems of Partial Differential Equations 10 du a 9 I xn 9u -1- In it -L tt F n ti gi m dt xn dx dx J dw b d dwy I xn 1 cw In w wg x ukwm . dt xn dx dx . Multiplicative separable solution u exp Amect y x w exp -Akect z x where A is an arbitrary constant and the functions y y x and z z x are determined by the system of ordinary differential equations ax n xn y x x cy In y yf x yk zm 0 bx -n xnz x X cz In z zg x yk zm 0. . Arbitrary functions depend on u2 w2. 11. du a 9 1 n du xn dx J uf u2 w2 - wg u2 w2 dt xn dx dw a d n dwy xn dx j wf u2 w2 ug u2 w2 . dt xn dx Time-periodic solution u r x cos
