Question

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 = 0

m =

Answer 1

Answer:

m = 6,9  

Step-by-step explanation:

We are given that $y = x^m$ is a solution to given differential equation.

$x^2y'' - 14xy' + 54y = 0$  

First we 3evaluate the value of:

$y'' = m(m-1)x^{(m-2)}$

$y' = mx^{(m-1)}$

Putting these value in the above differential equation, we get,

$x^2m(m-1)x^{(m-2)} + 14xmx^{(m-1)} + 54x^m = 0$

$x^m[m(m-1) -14m + 54] = 0$

$x^m(m^2 - 15m + 54) = 0$

$(m^2 - 15m + 54) = 0$

$(m-9)(m-6) = 0$

$m = 9,6$

Thus, for m = 9,6 , the function $y = x^m$ is a solution of given differential equation.