Question

Consider the differential equation X(1 — x)y" + [p – (p+ 2)x]y' – Py = 0, where p is a constant. (a) If p is not an integer, find the general solution near x = 0) in terms of Gauss's hypergeometric functions;

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Answer 1

Sa avem deterrendal equation is 24(1-2); +[P-P+2' - Py = 0 la (2) To solve the given equ² , we first knows that, -2 (1-2) Y" + [-Q4.4+) 2] y'- dßy = 0 where aßid are lenstand, the euro is known as hyper geometric or Gaun equation (bui v is not an integer). Then, solution of the hyper giometric elun (2) i y = a. 2k, Ca, ß; 8; 2) +62% 2F C1-87291-8+8; 2-8; 2) where a 6 are arbitri Erary constanta Ovad 2F (3, B;8; 2) or simply. F(,B; V; 2) is called hypergeometric function or hypergumenic rauss function is defined by z? 670 (0) Ź F(daß; 8; 2) an 7! deti). ß(3+1) 1:270+) z = [+ or 2 + o Now, comporring @ with (2), we have napi Hatß = p+2, BSP these, solving Itat P Pt 2 odwill + pco oo (2-1) (6-1) = ot, Р When, aal, p =P =P L = P, ß= 1; Pod 2005 mo, O fali

leze:(when d=1, p= P.D= Here 8=P is not an integer. The general sucution of (2) is y= autbre where U (8.0; 7; 2) = F(1, PP:) It 1.Pxt 1. (1+1) P (P+) 7+ 1.2. P 1841) 2 1.P sit X+ 2?+23 + 8; 2) = 2 v=x 2Ę(1+0+0, !- V+R; 2-9; 2) 2'0F (2-P,!, 2-P52) 21-P (1+2+2'+.--) 3-2 - 2-P +7 + 2 is +x 2-P +2 3-p t.- given by, g Hence, the general solution of a in this case = a ( 1+x+x?to--) +6 (2'-Pt -) Cesse : 11 ( when dap, p=1, 8=P) In this case we also get the senue generul socision as in cense (1). Then, the general solution of O near the pt. x=0 - is given by PF (2-P, 1; 2-P ; a) y=ar (1, P; p; x) + bx-P. -]+ balap = a[i+a+2²+x3 [1+2+3+4723+ ---) ; where a base arbitrary constent.