Question

Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y = 0, in the neighbourhood of the origin. a) Show that x = 0 is a regular singular point of the ODE. (10) b) By seeking an appropriate solution to the ODE, show that G=- (10) i) the roots to the indicial equation of the ODE are 0 and 1/2. [10] ii) the recurrence formula used to determine the power series coefficients, ens when one of the indicial equation roots is 1/2 is given by 7.00 (28+7) Cs+1 s = 1,2,..... (28+3)(8+1) iii) For the indicial equation root 1/2, show that one of the solutions to the ODE IS given by y = cova[1 -+ 21x2_77x (5) 40 560

Answer 1

For the indicial root n2 = ² (3 + ²) C. Co (1 ++) (2.4 +) Co 3 x 2 2 7 6 -C. for N2 = Į the the equation (viii) Cs becomes - ( + 5+3) Csel (+5+1) (2.4+25+1) ( 225+ (S+) (+1425 cs +7 2 Cs (25+3) (25+2) 2 - - (25+7) (25+3)(25+2) Cs S = 1,2,3, Ans

<b> To find the series solution of ti) about x=0, Let nes z con (c.+) s=0 Then we have x dy nes- dx = Z Cs (Mrs)x s=0 and э nts-2 Z as (nts) (nes-1) daz S=0 Substituting these values dy a dy i) in we have 2a I c (n+s) (n+S-1) nes-2 x S=0 8W (1 +*) nes-1 Cs (nts) x s=o 3 I can nes 4 = 0 S=0

co -(n+3) - C, (n+)( 2n+1) For s=2, (vi) gives (n+4) C2 = (1+2)( 2n + 3) S Ci 6- Co 61+4) - (213) (1+2) (2n+3) (-+) (2 M+)) (n+3) (new) (n-1) (21-3) (pi(2041) co