Question

Problem 5 For the equation given below, classify all singular points as regular or irregular. For those that are regular, compile indicial equations, find roots, and write theoretic necessary to solve the equations.] r +3 1-2-1-3 l).

Answer 1

X^3 - 5 x^2 + 6x = x (x ^2 - 5 x + 6) = x [x^2 - 3x - 2x + 6] = x [x (x - 3) phi - 2 (x - 3)) = x (x - 3) (x - 2) y'' + p (x) y' + Q(x) y = 0 P(x) = 3/x (x - 3) (x - 2)^2, Q (x) = negative 2 (x + 3)/ (x - 3)^2 (x - 2) lim_ x rightarrow 2 (x - 2) p (x) does not exists therefore x = 2 is an irregular singular point lim _x rightarrow 3 (x - 3) p(x) = lim _x rightarrow 3 3/x (x - 2)^2 = 3/3 = 1 = p_0 lim _x rightarrow 3 (x - 3)^2 p(x) = lim _x rightarrow 3 negative 2 (x + 2)/x (x - 2) = negative 10/3 = a_0 x = 3 is an regular singular point . \r\n the identical equation is given by r (r - 1) + p_0 r + a_0 = 0