Question

gnment Chapter 5, Section 5.6, Question 01 Consider the following differential equation. (a) Find all the regular singular points of the given differential equation If necessary, enter your answers separated by commas. page (b) Determine the indicial equation and the exponents at the singularity of each regular singular point Enter the values of ri and r2 in increasing order F(r)- ак a t

Answer 1

"9xy"" + 10xy' + 19e^x y = 0 implies y"" - 10/9 y' + 19/9 e^x y = 0 Here x = 0 is the only singular point, we show it is regular singular point. Here p(x) = 10/9 & q(x) - 19e^x/9x Since xp(x) = 10x/9 = x^2 theta (x) = 19xe^x/9 both are analytic at x = 0. Hence x = 0 is the only region singular point. Let y = sigma^infinity_n = 0 c_n x^n + 2 c_0 notequalto 0 y' = sigma^infinity_n = 0 c_n (n + r) x^n + r-1 y"" = sigma^infinity_n = 0 c_n (n + r) (n + r - 1) x^n + r-1 + 10 sigma^infinity_n = 0 c_n (n + r) x^n + r + 19 e^x sigma^infinity_n = 0 c_n x^n + r = 0 The least power of 'x' appearing in the equation is r - 1 when n = 0 equality it to zero, 9c_0 (r) (r - 1) = 0 implies r(r - 1) implies F� = r' - r & its roots are r = 0, 1 so r_1 = 0 & r_2 = 1"