The values of m for which $y = x ^ { m }$ is a solution of $x ^ { 2 } y ^ { \prime \prime } - 5 x y ^ { \prime } + 8 y = 0$ are
A) 2 and 4
B) $- 2$ and $- 4$
C) 3 and 5
D) 2 and 3
E) 1 and 5

Question

Quizplus · Accepted Answer

Given $y = x^m$, we find $y' = mx^{m-1}$ and $y'' = m(m-1)x^{m-2}$. Substituting these into the differential equation $x^2y'' - 5xy' + 8y = 0$ gives $x^2[m(m-1)x^{m-2}] - 5x[mx^{m-1}] + 8x^m = 0$. Simplifying, we get $m(m-1)x^m - 5mx^m + 8x^m = 0$, which simplifies further to $(m^2 - 6m + 8)x^m = 0$. Setting the quadratic equation $m^2 - 6m + 8 = 0$ equal to zero and solving for $m$ gives $m = 2$ and $m = 4$, which corresponds to choice A.

The Values of M for Which y=xmy = x ^ { m }y=xm

The Values of M for Which $y = x ^ { m }$