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Handbook of mathematics for engineers and scienteists part 72

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Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 72', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 12.1. First-Order Differential Equations 465 12.1.6-2. Reduction of the Abel equation of the second kind to the canonical form. 1 . The substitution w y g E where E exp - f2 dx 12.1.6.2 brings equation 12.1.6.1 to the simpler form ww x F1 x w F0 x 12.1.6.3 where F1 f1 - 2f2g g E Fo fo - fig f2g2 E2. 2 . In turn equation 12.1.6.3 can be reduced by the introduction of the new independent variable z F1 x dx 12.1.6.4 to the canonical form ww z - w R z . 12.1.6.5 Here the function R z is defined parametrically x is the parameter by the relations D Fo x r R z F1 x dx. F1 x Substitutions 12.1.6.2 and 12.1.6.4 which take the Abel equation to the canonical form are called canonical. Remark 1. The transformation w aW z az b brings 12.1.6.5 to a similar equation WWZ - W a-R az b . Therefore the function R z in the right-hand side of the Abel equation 12.1.6.5 can be identified with the two-parameter family of functions a-1R az b . Remark 2. Any Abel equations of the second kind related by linear in y transformations x 1 x y 2 x y Q 3 x have identical canonical forms up to the two-parameter family of functions specified in Remark 1 . 12.1.6- 3. Reduction to an Abel equation of the first kind. The substitution y g 1 u leads to an Abel equation of the first kind uX fo - fa f2g2 u3 f1 - 2f2g g x u2 f2u 0. For equations of this type see Subsection 12.1.5. 12.1.7. Equations Not Solved for the Derivative 12.1.7- 1. Method of integration by differentiation. In the general case a first-order equation not solved for the derivative F x y y x 0 12.1.7.1 can be rewritten in the equivalent form F x y t 0 t yx. 12.1.7.2 466 Ordinary Differential Equations We look for a solution in parametric form x x t y y t . In accordance with the first relation in 12.1.7.2 the differential of F is given by Fx dx Fy dy Ft dt 0. 12.1.7.3 Using the relation dy tdx we eliminate successively dy and dx from 12.1.7.3 . Asa result we obtain the system of two first-order ordinary differential equations F .