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Question B4 (20 marks) (a) Solve the following initial value problem by the power series method. Obtain the partial sum, s, only up to the x5 term, and evaluate the value at x1 = 0.5: dạy dy dx² + 3x +2y = 0, dx y(0) = 1, y'(0) = 1 (8 marks) (b) Evaluate the following integral using Simpson's rule (4 intervals) and Gauss quadrature (3 sampling points), and compare their errors. S*x* Inx dx (8 marks) Hints: Simpson's rule: n-1 -from Ax f(a) + f(b) +4 3 fi + 2 i=1 (first summation for odd numbers i and second summation for even numbers j) Gauss quadrature: b ከ a J= dx Ef scooter x 57 Žms (794444 - a wis = a + b + 2 a i=1 Number of points, n Sampling points, ši 0 Weights, Wi Number of points, n Sampling points, ši Weights, Wi 1 2 3 It 11 18 + V30 36 2 J 1 + 1 2 4 0 9 3 olu 100 5 It + + VN mi 18 - 130 36 ala It