Question

Solve the following differential equation by the Power Series Method term by term from a, to a4 y”-(x-5)y'+xy=0

Answer 1

solution: The given equation can be written dy (x-5) de + xy = 0 02 da? o comporting with dy dhe e pins de la Realy zo analytie af 420 an ordinary point. To We take y= no n- x dre nal .: PC) - -(x - 5) and &(n) = x Obviously PCW) and 6 (n) are Therefore a=0 in obtain the series solution of the equation aotaxtaxt Ê anan Therefore, dy e nan dy e n(n-1) and Putting the values of dure and given equation n(n-1) ank -(*-5) me and so on, Å n(n-1) an 2"-2_n ank" +52 nana n-2 and dx² ho2 in the dx we have n-2 n-1 n ana + ho ha! n- hai ha Intl O + ana nad e ChriDeine Inan nao nal an, (n+2) (n+1) anas? BWA ant O + als na2 na2 0 an, 29+ 693x 7 8 (n+2) (n+1) antz a"-ap - nana +594 +10 922 À 5 2 Chrisantian + Qox to any on = 292 +599 +(6az-a talo + 10 a)a + Î [(n+2)(n+1) anez .nan+5(4+) and it and 120 na2 an, nal

Since @b an identity, equating team and the coefficient of Powers of a to zero, we get the Constant vorrions .. on, 292 +594 = 0 on, 02 돌에 Gaz aq tao+1092=0 6a3 - a + 25 a az (2601 – 20) (1+2) (n+1) antz -nant (5075) and, tani=0 an-, - nan + Conta) ant), forena, an and on, antz = (172) (ml) Putting na as - 292 + 1503 12 12 01 12 12 ose me aq= 2y + 504 + (262, - Co) 04 (6765) - (1-1) the ralues ao, ai, az, az, as, Putting y = aut ane Q22² + a, & aant- - I aqx+5 (2604-09) art + t2(7194-200 20.1-1.2 ) + Qy (n - Ext to x + 71 24+---.) which in the required solutions where arbitreary -Constanta 2 +0,- as an a ane