You are given that
y=x^12 is a solution of the homogeneous differential equation
x^2\frac{d^2y}{dx^2}-9x\frac{dy}{dx}-24y=0 \text{.}
Now use the method of variation of parameters by setting
y=x^{12}v(x),
for an unknown function
v(x) to be determined, and find the complete general solution of
x^2\frac{d^2y}{dx^2}-9x\frac{dy}{dx}-24y=-6x^{-8}
\text{,}
in the form
y=Ax^{12}+Bx^β+y_{PI}(x), where
A and
B are arbitrary constants
and
y_{PI}(x) is a particular integral.