Download e-book for kindle: Advanced Mathematical Methods in Science and Engineering by S.I. Hayek

By S.I. Hayek

A set of an intensive variety of mathematical themes right into a plenary reference/textbook for fixing mathematical and engineering difficulties. subject matters coated comprise asymptotic tools, a proof of Green's services for traditional and partial differential equations for unbounded and bounded media, and extra.

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First differentiate eq. 15) partially with ~Ly = L °aY(X’o-) : ao[2(a - ao) + (a- ao)2 logx] ~r-2 where alo ~d gx xa=x If O-= O-o,then: --o L 00- do-=o-0 Thus, thesecond solution satisfying thehomogeneous differential equation isgiven by: y (x) Using the form of the Frobenius solution: y(x,o-)= xa + Ean(°’) xn +a n=l then differentiating the expressionfor y(x,~) with ff results in: oo ay(x,cr)a =a0x~ log x + Ea~(~) n÷~ +an(o ) xn+~ lo gx n=l n=l SERIES SOLUTIONS --logx OF ORDINARY Xan(o) xn÷° n=O DIFFERENTIAL 31 EQS.

Where6ao wasset equalto 1. 20) oneobtains: oo n=2 , x=-x YzCX)=EahC-l) xn-l+ E[(~+l)a~(~)] n=O n=3 + logx E[(O+ 1) an(O)]Oxn-1 n=3 Substitutingfor an(o) andperforming differentiationwith o results in: a° (o - 1) ~r(cr+ 2) ... (or +n - 2)(~r+ 2)(~r+ 3) ... 2..... 3..... n 2 =(n a0 -a° (o - 1) 0(0+ 2)... (o + n - 2)(0 + 2)(0+ 3) ... 2..... L-’~+g(n-3)+g(n) oo xn_l + 3 n, (--h-S-_3) t ½logXn~= which,uponshifting the indicesin the infinite series gives: n=3,4,5 .... - SERIES SOLUTIONS Y2(X)=X-I 2 OF ORDINARY DIFFERENTIAL xn+2 [ __3 2 1 ~_~ n!

2 2. Obtainthe solution to the followingdifferential equations, valid near x = 0. zdY4xy=O (a) d2y dx-~-+x d--~ " d:Y dy_y=O (c) dx~- - x dx 2 +2) y=0 (d) d2---~-Y-4xdY-(x dx2 dx x dZy dy , (e) d-~-- X~xx -tx+2) y (f) (g) (x2+l (i) (x-l) d3-~’Y + x2 d2-~Y+ 3xdy + dx 3 dx 2 dx +6xdY +6y=0 dx d~Y +y d2y dy (j) x d---X-T- d--~ + 4x3y= Y= 0 SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQS. 3 4. Classify all the finite singularities, if any, of the followingdifferential equations: (a) d2y d--~- +dyX~x+(x:Z-4) y = 2 (b) x2 d~Y +(l+x) dY +y = dx dx" 2 (c) (1- x21 d~ --2x dY +6y x ~ dx" dx (d’) x(1- x2~2d~Y- 2x d-~-Y+ (1+ x)Zy= !

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Advanced Mathematical Methods in Science and Engineering by S.I. Hayek

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