Answer Happy

3. Q[3] 20 points (a) 08 pts i. 04 pts Determine the singular points of the Legendre equation (1 - x?)y" - 2xy + a(a +1)y = 0 and give the smallest interval on which a series solution about x = 0 converges ii. 04 pts Write down the form a series solution about 30 = -1/2 and the smallest interval on which a series solution converges for the equation (1 + x?)y" + 2xy + 4x?y=0 (b) 12 pts Consider the Airy's differential equation /" - xy = 0, -~< x < 0. i. 04 pts Show that every point is an ordinary point and that if we look for a solution in the form of a power series about 20 = 0), c3 = 0 and the recurrence relation is given by (k + 2)(k+ 1)(x+2 – CX-1 = 0, h = 1, 2, 3, ... ii. 08 pts Deduce that the series solution of the given equation about to = 0 give by y(x) = coyl(x) + 142() where .-31 + 2.3 2.3.5.6 2.3...(3n-1)(3n) and 7 3+1 3.4.6.7 3.4...(3n)(3n+1) + + HER en(x) = (1 =[v+ + + . + 3.4