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i. 04 pts Determine the singular points of the Legendre equation (1 – ?)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 + 2?)y" + 2xy + 4x²y = 0 (b) 12 pts Consider the Airy's differential equation y" - xy = 0, -«<< < 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 Xo = 0, C2 = 0 and the recurrence relation is given by (k+2)(k + 1)x+2 - Ck-1 = 0, k = 1, 2, 3, ... ii. 08 pts Deduce that the series solution of the given equation about 20 = 0 give by y (30) = coy (0) + C1920) where 26 y(x) = 1 + + 2.3 + ... + ) 2.3.5.6 2.3...(3n-1)(3n) and 1962) (2+ y2(x) = x+ + 3.4.6.7 + 2-3n+1 3.4...(3n)(3n+1) 3.4 2