tailieunhanh - Handbook of mathematics for engineers and scienteists part 178

Handbook of mathematics for engineers and scienteists part 178. 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. | Chapter T5 Ordinary Differential Equations . First-Order Equations In this Section we shall often use the term solution to mean general solution. 1- yX f y - Autonomous equation. Solution x C. J f y Particular solutions y Ak where the Ak are roots of the algebraic transcendental equation f Ak 0. 2- yX f xMy - Separable equation. Solution y f x dx C. J g y J Particular solutions y Ak where the Ak are roots of the algebraic transcendental equation g Ak 0. 3- g x yX fi x y fo x - Linear equation. Solution y CeF eF e F f0X dx where F x f1 x dx. J g x J g x 4- 9 x y x fi x y fo x yk- Bernoulli equation. Here k is an arbitrary number. For k 1 the substitution w x y1-k leads to a linear equation g x w x 1 - k f1 x w 1 - k f0 x . Solution y1-k CeF 1 - k eF e F fo x dx where F x 1 - k f1 x dx. J g x 7 g x 5- yX f y x - Homogeneous equation. The substitution u x y x leads to a separable equation xu x f u - u. 1207 1208 Ordinary Differential Equations 6. yX ay2 bxn. Special Riccati equation n is an arbitrary number. 1 . Solution for n -2 1 w x a w w x x Ci J1 Vabxk C2Y1 Vabxk L 2k k 2 2k k where k 2 n 2 Jm z and Ym z are Bessel functions see Subsection . 2 . Solution for n -2 y A - xW- -x C - x 2aA 1 where A is a root of the quadratic equation aA2 A b 0. 7. yX y2 f x y - a2- af x . Particular solution y0 a. The general solution can be obtained by formulas given in Item 1 of equation . 8. yX f x y2 ay - ab - b2f x . Particular solution y0 b. The general solution can be obtained by formulas given in Item 1 of equation . 9. yX y2 xf x y f x . Particular solution y0 -1 x. The general solution can be obtained by formulas given in Item 1 of equation . 10. yX f x y2 - axnf x y anxn-1. Particular solution y0 axn. The general solution can be obtained by formulas given in Item 1 of equation . 11. yX f x y2 anxn 1 - a2x2nf x . Particular solution y0 axn. The general solution can be obtained by formulas given in Item 1 of equation . 12. yX - n