tailieunhanh - Handbook of mathematics for engineers and scienteists part 179

Handbook of mathematics for engineers and scienteists part 179. 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. | 1214 Ordinary Differential Equations 5. yXx ax b y x ax2 dx Y y 0. The substitution y u exp sx2 where s is a root of the quadratic equation 4s2 2as a 0 leads to an equation of the uXx a 4s x b u x 2bs x y 2s u 0. 6. xy x ay x by 0 1 . The solution is expressed in terms of Bessel functions and modified Bessel functions J C1JV 2VbX C2Yv 2VbX if bx 0 C1IV 2 bX C2K 2 bX if bx 0 where v 1 - a . 2 . For a 2n 1 where n 0 1 . the solution is C cos 4bx C2 sin Y4bx if bx 0 1 dxn 2 dxn C cosh 4 bx C2 sinh 4 bx if bx 0. dxn dxn 7. xyXx ayX bxy 0. 1 . The solution is expressed in terms of Bessel functions and modified Bessel functions 1-a _ _ x C1 J Vbx C2Y Vbx if b 0 1-a x C1 I V b x C2K bix if b 0 where v 1 - a . 2 . For a 2n where n 1 2 . the solution is n 1 d Y I r 1 d n i AO I Ci cosWo C2 smWb y x dx x dx 1 d n U 1 d Y1 u CTt I Cl t- cosh xv-b C2 sinh xv-b x dx x dx if b 0 if b 0. 8. xy X ny x bx1 2ny 0. For n 1 this is the Euler equation . For n 1 the solution is y C1 sin x1-n C2 co x1-n n-1 n-1 C1 exp i x1 C2 exp x1 n-1 n-1 if b 0 if b 0. 9. xyxx ay x bxny 0. If n -1 and b 0 we have the Euler equation . If n -1 and b 0 the solution is expressed in terms of Bessel functions y n l 2 b ra l x 2 C2YJ x 2 n 1 2Vb n 1 1 -a where v -----. n 1 . Second-Order Linear Equations 1215 10- yXx b - x y x - ay . Degenerate hypergeometric equation. 1 . If b 0 -1 -2 -3 . Kummer s series is a particular solution a b x I . i b kk where a a a 1 . a k - 1 a 0 1. If b a 0 this solution can be written in terms of a definite integral b t r b i 1 _xtta-1 1 t b-a-1 dt a b x r a r b-a J0 et 1 -t where V z J e f tz 1 dt is the gamma function. If b is not an integer then the general solution has the form y C1 a b x C2xl b a -b 1 2 - b x . 2 . For b 0 -1 -2 -3 . the general solution of the degenerate hypergeometric equation can be written in the form y C1 a b x C2 a b x while for b 0 -1 -2 -3 . it can be represented as y x1 -b C1 a-b 1 2 -b x C2 a-b 1 2 -b x . The .