tailieunhanh - Handbook of mathematics for engineers and scienteists part 181

Handbook of mathematics for engineers and scienteists part 181. 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. | 1228 Ordinary Differential Equations 28 yXx a yX 2 eßXf y ß vX Solution eay dy 1 f . F C1 C2 ß where F y J eayf y dy. 29 y L f y yX 2 g y The substitution w y y x 2 leads to a first-order linear equation w y 2f y w 2g y 0. 3 . y L f y yX 2-1 yX eXg y . The substitution w y e x y x 2 leads to a first-order linear equation w y 2f y w 2g y . 31 yXx xf y yX 3. Taking y to be the independent variable we obtain a linear equation with respect to x x y x Xjy -f y x. 32. y XXx f y yX 2 g x yX Dividing by y x we obtain an exact differential equation. Its solution follows from the equation ln yXI f y dy I g x dx C. Solving the latter for y x we arrive at a separable equation. In addition y C1 is a singular solution with C1 being an arbitrary constant. 33. y XXx f x g xyX- y The substitution w xy X - y leads to a first-order separable equation w x xf x g w . 34. y x yf X . The substitution w x xy x y leadstoa first-order separable equation xw x f w w-w2. 35- gyXX 2gXyX f y h XVg g g x - The substitution w y y x g leads to a first-order separable equation ww y f y h w . 36- y L f yx ay The substitution w y y x 2 ay leads to a first-order separable equation w y 2f w a. References for Chapter T5 Kamke E. Differentialgleichungen Lösungsmethoden und Lösungen I Gewöhnliche Differentialgleichungen B. G. Teubner Leipzig 1977. Murphy G. M. Ordinary Differential Equations and Their Solutions D. Van Nostrand New York 1960. Polyanin A. D. and Zaitsev V. F. Handbook of Exact Solutions for Ordinary Differential Equations 2nd Edition Chapman Hall CRC Press Boca Raton 2003. Zaitsev V. F. and Polyanin A. D. Discrete-Group Methods for Integrating Equations of Nonlinear Mechanics CRC Press Boca Raton 1994. Chapter T6 Systems of Ordinary Differential Equations . Linear Systems of Two Equations . Systems of First-Order Equations 1. x t ax by y t cx dy. System of two constant-coefficient first-order linear homogeneous differential equations. Let us write out the characteristic equation A2