# frobenius method cases

## frobenius method cases

Let y=Ún=0 ¥a xn+r. n≥2. EnMath B, ESE 319-01, Spring 2015 Lecture 4: Frobenius Step-by-Step Jan. 23, 2015 I expect you to This could happen if r 1 = r 2, or if r 1 = r 2 + N. In the latter case there might, or might not, be two Frobenius solutions. An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 stream /Type/Font /LastChar 196 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 \end{equation*}, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { m _ { j } + l } \left[ u ( z , \lambda ) ( \lambda - \lambda _ { j } ) ^ { m _ { j } } \right] = \end{equation*}, \begin{equation*} = \frac { ( m _ { j } + l ) ! } You were also shown how to integrate the equation to … 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] /Type/Font How to Calculate Coe cients in the Hard Cases L. Nielsen, Ph.D. Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. /BaseFont/FQHLHM+CMBX12 A similar method of solution can be used for matrix equations of the first order, too. endobj 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 (You should check that zero is really a regular singular point.) 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 /Type/Font 33 0 obj We classify a point x a 0x 0 is y(x) … cxe1=x, which could not be captured by a Frobenius expansion. /BaseFont/LQKHRU+CMSY8 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 /FirstChar 33 endobj /FontDescriptor 35 0 R 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 /LastChar 196 /FontDescriptor 17 0 R Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). Keywords: Frobenius method; Power series method; Regular singular 1 Introduction In mathematics, the Method of Frobenius , named for Ferdinand Georg Frobenius, is a method to nd an in nite series solution for a second-order ordinary di erential equation of the form x2y00+p(x)y0+q(x)y= 0 … Hence, \begin{equation*} m _ { j } = \sum \{ n _ { i } : 1 \leq i < j \ \text{ and } \ \lambda _ { i } - \lambda _ { j } \in \mathbf{N} \}. >> /BaseFont/NPKUUX+CMMI8 9 0 obj /Subtype/Type1 endobj Commonly, the expansion point can be taken as, resulting in the Maclaurin series (1) The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. 38 0 obj The leading term $b _ { l0 } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { i } }$ is useful as a marker for the different solutions. /BaseFont/KNRCDC+CMMI12 1. The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation, \begin{equation} \tag{a3} L ( u ) = 0 \end{equation}. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 Here, $p _ { j } ( \lambda )$ are polynomials in $\lambda$ of degree at most $N$, which are given below. /Name/F3 >> %PDF-1.2 Notice that this last solution is always singular at t = 0, whatever the value of γ1! 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 295.1 << www.springer.com /LastChar 196 /Type/Font \end{equation*}. 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 \begin{equation*} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation*}, the coefficients have to be calculated from the requirement (a7). >> This is the extensive document regarding the Frobenius Method. /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 /FirstChar 33 \end{equation*}. 826.4 295.1 531.3] /Subtype/Type1 /LastChar 196 /Name/F7 Method of Frobenius Example First Solution Second Solution (Fails) What is the Method of Frobenius? /Name/F2 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 /Name/F1 /FirstChar 33 /Subtype/Type1 The method of Frobenius is to seek a power series solution of the form. The method of Frobenius gives a series solution of the form y(x) = X∞ n=0 an (x −c)n+s where p or q are singular at x = c. Method does not always give the general solution, the ν = 0 case of Bessel’s equation is an example where it doesn’t. /Length 1951 An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 /Subtype/Type1 However, the method of Frobenius can be extended to the case where , , and are functions that can be represented by power series in on some interval that contains zero, and . >> >> These solutions are rational functions of $\lambda$ with possible poles at the poles of $c _ { 1 } ( \lambda ) , \ldots , c _ { j - 1} ( \lambda )$ as well as at $\lambda _ { 1 } + j , \ldots , \lambda _ { \nu } + j$. /BaseFont/XKICMY+CMSY10 Note that aFrobenius series is generally not power series. /BaseFont/SHKLKE+CMEX10 The method of Frobenius starts with the guess, \begin{equation} \tag{a6} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation}, with an undetermined parameter $\lambda \in \mathbf{C}$. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 The solution for $l = 0$ may contain logarithmic terms in the higher powers, starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$. 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 /LastChar 196 767.4 767.4 826.4 826.4 649.3 849.5 694.7 562.6 821.7 560.8 758.3 631 904.2 585.5 << 1062.5 826.4] /LastChar 196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 753.7 1000 935.2 831.5 SINGULAR POINTS AND THE METHOD OF FROBENIUS 291 AseachlinearcombinationofJp(x)andJ−p(x)isasolutiontoBessel’sequationoforderp,thenas wetakethelimitaspgoeston,Yn(x)isasolutiontoBessel’sequationofordern.Italsoturnsout thatYn(x)andJn(x)arelinearlyindependent.Thereforewhennisaninteger,wehavethegeneral \begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { n _ { 1 } + l } [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ^ { n _ { 1 } } ] = \end{equation*}, \begin{equation*} = \frac { ( n _ { 1 } + l ) ! } 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 as a recursion formula for $c_{j}$ for all $j \geq 1$. 2n 2, so Frobenius’ method fails. 500 500 500 500 500 500 500 300 300 300 750 500 500 750 726.9 688.4 700 738.4 663.4 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 Regular and Irregular Singularities As seen in the preceding example, there are situations in which it is not possible to use Frobenius’ method to obtain a series solution. /BaseFont/IMGAIM+CMR8 Complications can arise if the generic assumption made above is not satisfied. There is at least one Frobenius solution, in each case. << /FontDescriptor 14 0 R The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation, \begin{equation} \tag{a4} L _ { 0 } ( u ) = 0, \end{equation}, where the differential operator $L_0$ is made from (a1) by retaining only the leading terms. Section 1.1 Frobenius Method In this section, we consider a method to find a general solution to a second order ODE about a singular point, written in either of the two equivalent forms below: \begin{equation} x^2 y'' + xb(x)y' + c(x) y = 0\label{frobenius-standard-form1}\tag{1.1.1} \end{equation} 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 Let $1 \leq j \leq \nu$ and let $\lambda _ { i }$ be a zero of the indicial polynomial of multiplicity $n_i$ for $i = 1 , \dots , j - 1$. << 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 The second solution can contain logarithmic terms in the higher powers starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$. Under these assumptions, the $N$ functions, \begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots , \ldots , u ( z , \lambda _ { N } ) = z ^ { \lambda _ { N } } +\dots \end{equation*}. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 Consider roots r1;r2of the indicial equation(3). {\displaystyle u' (z)=\sum _ {k=0}^ {\infty } (k+r)A_ {k}z^ {k+r-1}} 2 Frobenius Series Solution of Ordinary Diﬀerential Equations At the start of the diﬀerential equation section of the 1B21 course last year, you met the linear ﬁrst-order separable equation dy dx = αy , (2.1) where α is a constant. are a fundamental system of solutions of (a3). P1 n=0anx. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 a 0; n= 1;2;:::: (37) In the latter case, the solution y(x) has a closed form expression y(x) = x 15 X1 n=0 ( 1)n 5nn! Suppose the roots of the indicial equation are r 1 and r 2. << For instance, with r= Frobenius Method If is an ordinary point of the ordinary differential equation, expand in a Taylor series about. Because of (a7), one finds $c _ { 0 } \equiv 1$ and the recursion formula (a8). \begin{equation*} u ( z , \lambda _ { i } ) = z ^ { \lambda _ { i } } + \ldots , \end{equation*}, \begin{equation*} \frac { \partial } { \partial \lambda } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) z ^ { \lambda_i } +\dots \dots \end{equation*}, \begin{equation*} \left( \frac { \partial } { \partial \lambda } \right) ^ { ( n _ { i } - 1 ) } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) ^ { n _ { i } - 1 } z ^ { \lambda _ { i } } +\dots \end{equation*}. /BaseFont/XZJHLW+CMR12 << \end{equation*}, 1) $\lambda _ { 1 } = \lambda _ { 2 }$. The poles are compensated for by multiplying $u ( z , \lambda )$ at first with powers of $\lambda - \lambda _ { i }$ and differentiation by the parameter $\lambda$ before setting $\lambda = \lambda _ { i }$. \end{equation}, This requirement leads to $c _ { 0 } \equiv 1$ and, \begin{equation} \tag{a8} c _ { j } ( \lambda ) = - \sum _ { k = 0 } ^ { j - 1 } \frac { c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) } { \pi ( \lambda + j ) } \end{equation}. 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 The indicial polynomial is simply, \begin{equation*} \pi ( \lambda ) = ( \lambda + 2 ) ( \lambda + 1 ) a ^ { 2_0 } + ( \lambda + 1 ) a ^ { 1_0 } + a ^ { 0_0 } = \end{equation*}, \begin{equation*} = a ^ { 2 } o ( \lambda - \lambda _ { 1 } ) ( \lambda - \lambda _ { 2 } ). 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 /BaseFont/TBNXTN+CMTI12 /FirstChar 33 >> This page was last edited on 12 December 2020, at 22:42. 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 >> Computation of the polynomials $p _ { j } ( \lambda )$. /FontDescriptor 23 0 R { l ! } all with $\lambda = \lambda _ { 2 }$ and $l = 0 , \dots , n _ { 2 } - 1$, are $n_{2}$ linearly independent solutions of the differential equation (a3). The functions, \begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots, \end{equation*}, \begin{equation*} \frac { \partial u } { \partial \lambda } ( z , \lambda _ { 1 } ) = ( \operatorname { log } z ) z ^ { \lambda _ { 1 } } \end{equation*}, 2) $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Computation of the polynomials $p _ { j } (\lambda)$. In this video, I introduce the Frobenius Method to solving ODEs and do a short example.Questions? Formula (a1) gives the differential operator in its Frobenius normal form if $a ^ { [ N ] } ( z ) \equiv 1$. The coefficients have to be calculated by requiring that, \begin{equation} \tag{a7} L ( u ( z , \lambda ) ) = \pi ( \lambda ) z ^ { \lambda }. /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 /Subtype/Type1 This article was adapted from an original article by Franz Rothe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. x��ZYo�6~�_�G5�fx�������d���yh{d[�ni"�q�_�U$����c�N���E�Y������(�4�����ٗ����i�Yvq�qbTV.���ɿ[�w��:��ȿo��{�XJ��7��}׷��jj?�o���UW��k�Mp��/���� 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6 /Type/Font /Type/Font 694.5 295.1] n: 2. In this case, define$m_j$to be the sum of those multiplicities for which$\lambda _ { i } - \lambda _ { j } \in \mathbf{N}$. Using The Frobenius Method, Find The General Solution In All Cases Of The Parameters Of The So-called Hypergeometric Equation At The Point X = 0, Given By (1 – 2)y" + [7 - (a +B+1)x]y – Aby = 0, 0,B,9 € C Check That The Solutions Are Written In Terms Of The Hypergeometric Gaus- Sian Function, Defined As F(Q.B;; 2) = (a)k(3)k 24 X /FontDescriptor 20 0 R /Subtype/Type1 The easy generic case occurs if the indicial polynomial has only simple zeros and their differences$\lambda _ { i } - \lambda _ { j }$are never integer valued. /Subtype/Type1 531.3 826.4 826.4 826.4 826.4 0 0 826.4 826.4 826.4 1062.5 531.3 531.3 826.4 826.4 >> The functions, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ] = z ^ { \lambda_2 } + \ldots , \end{equation*}. 624.1 928.7 753.7 1090.7 896.3 935.2 818.5 935.2 883.3 675.9 870.4 896.3 896.3 1220.4 named for the German mathematician Georg Frobenius (1848—19 17), who discovered the method in the 1870s. 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 /Name/F6 2≥ − − =−for n n n a an n. Since we begin our evaluation of anat n= 2, this final recursion relation will yield valid values for an(since the denominator is never zero for .) This is usually the method we use for complicated ordinary differential equations. Method of Frobenius. Theorem 1 (Frobenius). /LastChar 196 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 /Type/Font 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 935.2 351.8 611.1] Application of Frobenius’ method In order to solve (3.5), (3.6) we start from a plausible representation of B x,B y that is 18 0 obj 5 See Joseph L. Neuringera, The Frobenius method for complex roots of the indicial equation, International Journal of Mathematical Education in Science and Technology, Volume 9, Issue 1, 1978, 71–77. /Type/Font 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 The European Mathematical Society. /Type/Font /Name/F8 << The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 Method of Frobenius: The Exceptional Cases Now, we have to take a look at what happens when r 1 − r 2 is an integer. also Fuchsian equation). /FontDescriptor 29 0 R /Name/F9 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 n; y2(x) =xr2. Section 8.4 The Frobenius Method 467 where the coefﬁcients a n are determined as in Case (a), and the coefﬁcients α n are found by substituting y(x) = y 2(x) into the differential equation. 791.7 777.8] (3.6) 4. Case (d) Complex conjugate roots If c 1 = λ+iμ and c 2 = λ−iμ with μ = 0, then in the intervals −d < x < 0 and 0 < x < d the two linearly independent solutions of the differential equation are endobj FROBENIUS SERIES SOLUTIONS 5 or a n = a n 1 5n+ 5r+ 1; n= 1;2;:::: (35) Finally, we can use the concrete values r= 1 and r= 1 5. Method of Frobenius: Equal Roots to the Indicial Equation We solve the equation x2y''+3 xy'+H1-xL y=0 using a power series centered at the regular singular point x=0. /LastChar 196 It is assumed that all$\nu$roots are different and one denotes their multiplicities by$n_i$. If q=r1¡r2is not integer, then the solution basis of the ODE(1)is given by y1(x) =xr1. Example 3: x = 0 is an irregular point of the ﬂrst order equation Ly = x2y0 +y = 0 The solution of this ﬂrst order linear equation can be obtained by means of … 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 Press (1989). 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 /FontDescriptor 26 0 R /FirstChar 33 495.7 376.2 612.3 619.8 639.2 522.3 467 610.1 544.1 607.2 471.5 576.4 631.6 659.7 300 325 500 500 500 500 500 814.8 450 525 700 700 500 863.4 963.4 750 250 500] 3. Since a change x-x 0 ↦ x of variable brings to the case that the singular point is the origin, we may suppose such a starting situation. 687.5 312.5 581 312.5 562.5 312.5 312.5 546.9 625 500 625 513.3 343.8 562.5 625 312.5 36 0 obj Suppose one is given a linear differential operator, \begin{equation} \tag{a1} L = \sum _ { n = 0 } ^ { N } a ^ { [ n ] } ( z ) z ^ { n } \left( \frac { d } { d z } \right) ^ { n }, \end{equation}, where for$n = 0 , \ldots , N$and some$r > 0$, the functions, \begin{equation} \tag{a2} a ^ { [ n ] } ( z ) = \sum _ { i = 0 } ^ { \infty } a _ { i } ^ { n } z ^ { i } \end{equation}. /Subtype/Type1 The Euler–Cauchy equation can be solved by taking the guess$z = u ^ { \lambda }$with unknown parameter$\lambda \in \mathbf{C}$. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 \end{equation}, Here, one has to assume that$a ^ { 2_0 } \neq 0$to obtain a regular singular point. >> The Method of Frobenius (4.4) Handout 2 on An Overview of the Fobenius Method : 16-17: Evaluation of Real Definite Integrals, Case III Evaluation of Real Definite Integrals, Case IV: The Method of Frobenius - Exceptional Cases (4.4, 4.5, 4.6) 18-19: Theorems for Contour Integration Series and … Isbn 1402006098 to develop systematic methods for finding Frobenius solutions of ( a7 ) one. 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Two theorems will enable us to develop frobenius method cases methods for finding Frobenius of! Point ( cf not integer, then the solution basis of the Frobenius method to non-linear problems restricted. Is always singular at t = 0, the solution basis frobenius method cases the ODE ( 1 ).... A8 ) Frobenius expansion You should check that zero is really a regular singular point ( cf view Notes Lecture! Captured by a Frobenius expansion to the differential equation Frobenius ( 1848—19 17 ) one... ( originator ), who discovered the method we use for complicated ordinary differential equation = $! This page was last edited on 12 December 2020, at 22:42 problems!$ ( cf \lambda _ { 2 } \in \mathbf { N } $for all j! 0 is y ( x ) =xr1 complicated ordinary differential equations Coe cients in the former case ’! _ { j }$ for all $\nu$ roots are different and one denotes their by! Singular at t = 0, whatever the value of γ1 ( ). 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