tailieunhanh - Handbook of mathematics for engineers and scienteists part 182

Handbook of mathematics for engineers and scienteists part 182. 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. | . Linear Systems of Two Equations 1235 where C1 and C2 are arbitrary constants. On substituting 2 into 1 and integrating one arrives at the general solution of the original system in the form x u t dt y v t 2- dt t2 where C3 and C4 are arbitrary constants. 6. x t f t aix hr y y f t a2x b2y . Let ki and k2 be roots of the quadratic equation k2 - ai b2 k aib2 - a2bi 0. Then on multiplying the equations of the system by appropriate constants and on adding them together one can rewrite the system in the form of two independent equations z kif t zi zi a2X ki - ai y z2 k2f t z2 Z2 a2X k2 - ai y. Here a prime stands for a derivative with respect to t. 7. xtt f t arx t bry t y t f t a2xt b2y t . Let ki and k2 be roots of the quadratic equation k2 - ai b2 k aib2 - a2bi 0. Then on multiplying the equations of the system by appropriate constants and on adding them together one can reduce the system to two independent equations z kif t Zi zi a2x ki - ai y z 2 k2f t z 2 Z2 a2X k2 - ai y. Integrating these equations and returning to the original variables one arrives at a linear algebraic system for the unknowns x and y a2x ki - ai y C exp kiF t dt C2 a2x k2 - ai y C3 J exp k2F t dt C4 where Ci . C4 are arbitrary constants and F t f t dt. 8. xtt af t tyt-y y t bf t txt-x . The transformation u txt - x v tyt - y i leads to a system of first-order equations u t atf t v v t btf t u. 1236 Systems of Ordinary Differential Equations The general solution of this system is expressed as lu t C1 a expl v ab J tf t dt C2a expl-v ab J tf t dt if ab 0 I v t C1 Vab exp f Vab tf t dt - C2Vab exp f -Vab tf t dt J J _ if ab 0 2 u t C1 a cosi ab J tf t dt C2a sinf ab J tf t dt v t -C labl sin ab J tf t dt C23 ab cos y ab tf t dt where C1 and C2 are arbitrary constants. On substituting 2 into 1 and integrating one obtains the general solution of the original system x C3t t u t t2 dt y C4t t v t t2 dt where C3 and C4 are arbitrary constants. 9. t2Xtt a1tXt b-1ty t c1x d1y 0 t2ytt a2tx t b2ty t