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Handbook of mathematics for engineers and scienteists part 116. 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. | 15.14. Nonlinear Sy stems of Partial Differential Equations 773 A consistency condition for the system can be found by setting F -P R and G -Q R in 15.14.1.2 . After some rearrangements we obtain the equation R. - FL p FL - FL q - FP 0. 15.14.2.5 dy dx J dz dy J dx dz If condition 15.14.2.5 is satisfied identically the Pfaffian equation 15.14.2.1 is integrable by one relation of the form U x y z C 15.14.2.6 where C is an arbitrary constant. In this case the Pfaffian equation is said to be completely integrable. The left-hand side of a completely integrable Pfaffian equation 15.14.2.1 can be represented in the form P x y z dx Q x y z dy R x y z dz p x y z dU x y z where p x y z is an integrating factor. A solution of a completely integrable Pfaffian equation can be found by solving the overdetermined system 15.14.2.4 employing the method presented in Subsection 15.14.1. Alternatively the following equivalent technique can be used it is first assumed that x const in equation 15.14.2.1 which corresponds to dx 0. Then the resulting ordinary differential equation is solved for z z y where x is treated as a parameter and the constant of integration is regarded as an arbitrary function of x C p x . Finally by substituting the resulting solution into the original equation 15.14.2.1 one finds the function p x . Example 1. Consider the Pfaffian equation y xz a dx x y b dy x2ydz 0. 15.14.2.7 The integrability condition 15.14.2.5 is satisfied identically. Let us set dx 0 in equation 15.14.2.7 to obtain the ordinary differential equation y b dy xydz 0. 15.14.2.8 Treating x as a parameter we find the general solution of the separable equation 15.14.2.8 z -1 y b ln y Nx 15.14.2.9 x where x is the constant of integration dependent on x. On substituting 15.14.2.9 into the original equation 15.14.2.7 we arrive at a linear ordinary differential equation for x xd fx .rz a 0. Its general solution is expressed as x - ln x C xx where C is an arbitrary constant. equation 15.14.2.7 .
