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Date Created: 09/28/15
ENM 511 J L Bassani April 2 2001 Frobenius Solution to a 2quot 1 order ODE near a regular singular point Consider the ODE y x Px y x Qx yx 0 1 We will look for series solutions to 1 around at most regular singular points which without loss in generality will be located at x 0 The notation adopted below closely follows that in the notes of Carchidi handout and on web site and in Hildebrand Sec 44 7 with his Rxl multiplying y If functions Px and Qx are regular around x0 ie if they possess a Taylor series expansion around x0 then yx can also be expressed in a Taylor series around x0 Substituting the Taylor series for Px Qx and yx into 1 and requiring that each term of like powers of x sums to zero each coefficient of a Taylor series for zero is zero leads to the unknown coefficients of the series for yx The method of Frobenius is an extension of this idea to equations with regular singular points and builds on what we know about equidimensional equations see handout on equidimensional equations If 1 has a regular singular point at x 0 then the following limits exist p0 11mxPx and ac 11mx 2 x4gt0 x4gt0 Note that the existence of these limits implies that 1 can be rewritten in the form II p x I Q x y ltxgt y ltxgt 2yltxgto 3 x x where and are regular around x0 in which case 3 is a generalization of an equidimensional equation Given this latter observation and what we know about Taylor series solutions about regular points we look for Frobenius series solutions 1 or 3 yx yx s x5 2aquot xquot 2aquot xquot 4 n0 n0 The exponent s is determined from the quadratic since 1 is a 2quotd order ODE indicial equation fs32p0 lsq00 5 which leads to two values of 3 By convention we order 31 and 32 such that Resl gt Re32 In principle each root gives a Frobenius series we must be careful about special cases just as with equidimensional equations The coefficients of those series are given in equations 29 with 27b of Carchidi s notes where pquot and qquot are the Taylor coefficients of the respective series for xPx and x2 Next we summarize 3 special cases before we take advantage of Maple to do the tedious algebra One solution to 1 can always be expressed in the form of 4 W x51 2a xquot 2a x 6 n0 n0 The second solution is determined according the following 3 cases Case 1 Consider s1 32 not an integer including when 31 and 32 are complex conjugates 7 case 1 includes Carchidi s cases a and c y2 x JCS2 2b xquot 2b meZ 71 n0 n0 Case 2 Consider 31 32 E so Carchidi s case b y2 x W W 2bquot We 72 n0 Case 3 Consider 31 32 is a positive integer Carchidi s case b y2 x cy1x lnx 2b meZ 73 n0 where the constant c can be zero or nonzero The strategy to solve a particular problem is to rst determine SI and 32 from 5 and go on to nd the coef cients an for y1xin 6 by summing the coef cients of like powers of x to zero The second solution is found in a similar way using the form for y2 x given by cases 1 2 or 3 Two handwritten examples appear below One can also use Maple For example to determine Frobenius solutions to Bessel s equation of zeroth order the following code can be used De ne Bessel s equation of zeroth order gt L u x 2diff ux x2 xdiff ux x aA2xA2 ux 2 Lu x2 ux x ux a2 x2 ux gtassumea real agt0 Obtain a series solution for J 0ax and Y0ax gtOrder 7 dsolveLuOuxseries l l ux7C1 l Zawzx2 l N4 4 N6 6 7 64a x 2304a xOx 7C2 aN6 x6 Ox7 4 x4 1 2304 I l 22 l lnxl 4aN x 64aN 1 2 2 3 4 4 11 6 5 7 4aNx 128aNx13824aNxOxH Or Maple can solve the equation symbolically gt sol dsolve L u O sol ux 7C1 BesselJ0 aN x 7C2 BesselY0 aN x Although Maple has done tedious algebra for us in determining the Frobenius series one still should determine the pattern in the coef cients ie the recursion relation For example the series for Jquot x can be expressed compactly as Jquotx i icfz n k0 Handout of 12901 note for the equidimensional equation only 31 32 E so is a special case Equidimensional ODEs Euler equations have a regular singular point which in the following examples is located at x0 0 ie uNxS see pp 142143 in Wylie and Barrett 7 more on singular points when we discuss the method of Frobenius Since the exponent s can be negative the solution can be singular at x 0 Furthermore when an exponent s is a repeated root of the characteristic equation solutions also exist of the form u lnsm x5 m l 2 M l where M is the multiplicity ofthe root Examples for various cases follow 1 Lu xzu 2xu 2u 0 Lxs 3s l 2s2xs s 3s2xs s ls 2xs The characteristic equation s2 3s 2 0 has roots sl l and 32 2 gt ux clx 02x2 no singularity at x 0 2 Lu 2x2u 5xu u 0 Lxs 233 1 5s l x5 2s2 3s lxs 2s ls l x5 gt ux clxl2 czyc391 singularity at x 0 3 Lu xzu xu u 0 It ss l slxs s 25lxs s l2 xs Hence 5 l is a double root M 2 Note that Lxs 2s 1xS s l2lnx xs 0 ifsl S Since Lxs L a J it follows that axs as xlnx is also a solution 51 The complete solution is ux 01x clenx To see that this solution is not singular at x 0 note that lnx 11m x5 lnx 2 11m an an xs l x 2 11m 1 an Sxs x5 lim an S 0 if s gt 0
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