Find all values of m so that the function y = xm is a solution of the given differential equation. (Enter your answers as a comma-separated list.)x^2y'' − 14xy' + 54y = 0m =

Answer:m = 6,9   Step-by-step explanation:We are given that [tex]y = x^m[/tex] is a solution to given differential equation.[tex]x^2y'' - 14xy' + 54y = 0[/tex]  First we 3evaluate the value of:[tex]y'' = m(m-1)x^{(m-2)}[/tex] [tex]y' = mx^{(m-1)}[/tex]Putting these value in the above differential equation, we get,[tex]x^2m(m-1)x^{(m-2)} + 14xmx^{(m-1)} + 54x^m = 0[/tex][tex]x^m[m(m-1) -14m + 54] = 0[/tex] [tex]x^m(m^2 - 15m + 54) = 0[/tex][tex](m^2 - 15m + 54) = 0[/tex][tex](m-9)(m-6) = 0[/tex][tex]m = 9,6[/tex]Thus, for m = 9,6 , the function [tex]y = x^m[/tex] is a solution of given differential equation.