Handbook of mathematics for engineers and scienteists part 185

Handbook of mathematics for engineers and scienteists part 185. 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. | 1256 First-Order Partial Differential Equations 9. W w yg w h w 0. ox oy . xf w h w f w r z z z General solution y ----------1 ----- exp g w x w . g w g2 w 10 fWgWw o. ox oy f dy 7 General solution y - h w J f x dx w . . Equations of the Form dllF f x v w L a x u w dx f X y W dy g x y w In the solutions of equations z is an arbitrary composite function whose argument z can depend on x y and w. 1. 2W aw f x . ax ay General solution y ax w - F x a F x dx w - F x where F x f x dx. 2. TT awTT f y . ox oy General solution x d u where F y f y dy u F y - 1 aw2. yo y aF z 2au J 2 3. aw f x g x . ox oy General solution where 4. f w g x . ox rg General solution y y ax w - G x a J G x dx F x w - G x F x J f x dx G x J g x dx. 9w z . dy i f G t - G x w dt w - G x where G x Jxo . Quasilinear Equations 1257 5. f w g y . ox oy General solution t- y x G t - G y F w dt F w G y dy0 where G y J g y dy and F w J f w dw. The function y y z is defined parametrically by y y z F w . 6. it f w it g w . ox oy . . f w . f dw General solution y y - dw Ql x - y I. 7. TT f w g x TT h x . ox oy General solution y Î f h t - H x w dt G x w - H x 7x0 where y h t G x J g x dx H x J h x dx. 8 TT f w 5 x l h w ox oy General solution f f w ÎW g H t -H w x trz u tzz . f dw TT dw ------- ---------dt 1 x - H w J where H x -r . J h w JW0 h t y J h w d dw 9 dx f w ya x dy h x General solution yG x - J G x f H t - H x w dx w -H x where G x exP f- yg x j. . h x yh x dx. 10 f x w g x . dx ay fX General solution y f t G t -G x w dt w-G x where G x 7x0 1258 First-Order Partial Differential Equations 11 f x w g w . dx dy w f G t G w x t dw General solution y ----------------- dt x-G w where G w -----. Jw0 g t J g w . Nonlinear Equations . Equations Quadratic in One Derivative In this subsection only complete integrals are presented. In order to construct the corresponding general solution one should use the formulas of Subsection . 1. dw dw 2 dx dy y This equation .