Question

(1 point) Classify each singular point as regular (r or irregular (). (12-51-24)2x" + (t2-9)r,-u = 0 List the singular points in increasing order: is The singular point ti - IS is The singular point t2 - IS Which of the following statements correctly describes the behaviour of the solutions of the differential equation near the singular point A. All solutions remain bounded near t B. At least one non-zero solution remains bounded near t and at least one solution is unbounded near t C. All non-zero solutions are unbounded near t

Answer 1

(t^2 - 5t - 24)^2 x'' + (t^2 - 9)x' - tx = 0 singular points are roots of t^2 - 5t - 24 = 0 Roots are t = 8, - 3 t^2 - 5t - 24 = 0 or, t^2 -8t + 3t - 24 = 0 or, t(t - 8) + 3(t - 8) = 0 or, (t - 8)(t + 3) = 0 (t + 3) t^2 - 9/(t^2 - 5t - 24)^2 = t - 3/(t - 8_^2, (t + 3)^2(- t)/(t - 8)^2(t + 3)^2 = -t/(t - 8)^2 are analytic at t = -3 (t - 8) t^2 - 9/(t - 8)^2(t + 3)^2 = t^2 - 9/(t - 8)(t + 3) is not analytic at t = 8 Therefore t_2 = 8 is irregular singular point t_1 = -3 is regular singular print. (B) Atleast on non zero solution remains bounded near t_1 = -3 and at least one is unbounded.