By S.I. Hayek

A suite of an intensive diversity of mathematical themes right into a plenary reference/textbook for fixing mathematical and engineering difficulties. subject matters coated contain asymptotic equipment, an evidence of Green's services for traditional and partial differential equations for unbounded and bounded media, and extra.

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A suite of an intensive diversity of mathematical issues right into a plenary reference/textbook for fixing mathematical and engineering difficulties. subject matters lined contain asymptotic equipment, a proof of Green's features for traditional and partial differential equations for unbounded and bounded media, and extra.

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29), mustbe identical. 2 1. Solvethe followingdifferential equations: 2 (a) d-~Y+xy:e-X~/2 ax (c) (b) dY+2ycotanx=cosx dx ¯ dy (e) sin x cos x-- + y = sin dx Section 2. 4 3. 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.

7) W Jp,Yp JpYp = ~x ( ) = JpYp , - ,__2 whichis independentof p. 4) axe applied. Fromthe recurrenceformula,eq. 2), oneobtainsthe followingby setting p = am+2 = am m= 0, 1, 2 .... z(m+ a + 2) Again,by induction,onecan showthat the evenindexedcoefficients are: m a2m= (-1) at z (0- + 2)z(0-+ 2 .. (0- + 2m y(x,0-) at xa + atZ(-1)m m=1,2.... x2m+a . + 4)~ 2... (0- + 2m) (0- + 2)2(0 m=l Usingthe formfor the secondsolution given in eq. 16), oneobtains: y2(x)= °~Y(X’0-) = at xa logx +ao logx ~ c90- 0-0 = 0 m=1 -2a0 E(-I) m (0- + (-1)mx2m+a +2m)z (0-+2)2(a--’-~"~-~ 2)2(0.

20) It c~ ~ shown ~at ~e l~t infinite ~fies is pro~onal m y~(x). 6 Obtain the solutions of the followingdifferential equation about x0 = O: 2 9 Since xo -- 0 is a RSP,then substituting the Frobeniussolution into the differential equationresults in: n+ff)2 = - an xn+°-2+ ~’ a nn+ff X =0 ~_~ n=0 which, uponextracting the two terms with the lowest powersof x, gives: SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQS. 35 2 --~] alx°-I + (~r2 --~] a0x°-2 +[(o+1) m=0 Thus, equating the coefficient of each powerof x to zero; one obtains: and the recurrence formula: a,=*2= am (m+2+tr)=_9~ am = (m+a+~)(m+a+7//2) m=0, 1,2 ....