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Question 3 Consider the ordinary differential equation (ODE) 2xy" + (1 + x)y' + 3y =...


Question 3 Consider the ordinary differential equation (ODE) 2xy + (1 + x)y + 3y = 0, in the neighbourhood of the origin. a
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
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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 -<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 nco -(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+)) (

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