By S.I. Hayek

A set of an intensive diversity of mathematical issues right into a plenary reference/textbook for fixing mathematical and engineering difficulties. subject matters lined contain 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.

2m- n). (~ + 2 + n)... (~ + 2m m=1,2,3 .... 4) must nowbe followed. oo xa a0 y(x,~r) + a0 x2m+ ~"’ ~zLA-1)m (or + 2 - n)... n). ((r + 2 + n)... (~r m=l Then, the secondsolution for the case of an integer difference k = 2n is: ny2(x) : ~-{(~r-if2)y(x, ff)}o=a2 = ~{(ff + n)y(x,~)},___ Usingthe formulafor Y2(X)in eq. 20), an expressionfor Y2results: m=~ ~n- 1 y2(x) + a0 a0 x2m_ n zLA-1)m(2- 2n)(4- 2n)... (2m - 2n). 4-... m=0 :n ’(o + 2 - n) ... (o + 2m- n): ~-’+"~+ n) ... (o + 2m r mn) + ao logx ~ L!

Obtain the homogeneoussolution to the followingdifferential (a) d3y (b) 3 dx dzy dy 2y=0 dx z dx equations: d2y 1 dy dx 2 4 dx ÷ Y = 0 (c) d3Y-3dY+2y=0 dx d4y (d) d-’~-- 8 d2y ~x ~+ 16y=0 d4y (e) ~-T- 16y= d~y (f) d---~ + iy = 0 day (g) d--~- + 16y = dd~-~Y3 2 d~y + dy (h) dSY d4Y dx 2_.. + dx--- T ~--y=0 dx 4 5 (i) d3--~-Y+ 8a3y= 3dx d2Y + 2a3y = 0 d3Y (j) ~x 3 -T a dx--- d4y + 2 (k) ~-~- ~ +a4y=O (1) d~6Y+ 64y = 0 i = ~Z-~ CHAPTER 4. 1 16 If a third order differential equation,with constantcoefficients, has three repeated roots = m, showthat e mx, xe mx, and x2e mx makeup a linearly independentset.

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