3. Q[3] 20 points (a) 08 pts i. 04 pts Determine the singular points of the Legendre equation (1 - x?))" - 2xy +a(a + 1)

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3 Q 3 20 Points A 08 Pts I 04 Pts Determine The Singular Points Of The Legendre Equation 1 X 2xy A A 1 1 (38.01 KiB) Viewed 34 times

3. Q[3] 20 points (a) 08 pts i. 04 pts Determine the singular points of the Legendre equation (1 - x?))" - 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 so = -1/2 and the smallest interval on which a series solution converges for the equation (1 + x?)y' + 2xy +4.ry=0 (b) 12 pts Consider the Airy's differential equation y" - ry = 0, -<x<. 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 10 = 0, y = 0 and the recurrence relation is given by (k + 2)(2 + 1)x+2 -0-1 = 0, k = 1,2,3.... ii. 08 pts Deduce that the series solution of the given equation about 20 = 0 give by y(x) = coyl () + C142(X) where 3 3 2.3.5.6 2.3... (3n-1)(3n) + + 2.3 + + and 21 (ar) = [1 (m) [+5+ - v . r + 3.4 27 + 3.4.6.7 + 3+1 3.4...(3n)(3n+1)