I'm given a solution $y = at^n$ (fixed)
And then asked if there exists a second order linear homogeneous differential equation for it.
My current line of thinking is that I can just substitute the solution into the form of a DE and if the resulting equation "looks right", then it exists. However, it bothers me that I can't think of a solution that would be problematic.
This is how I'm doing it:
Solution is $y = 2/t$
$y' = 2 * (-1) * 1/t^2 = -2/t^2$
$y'' = -2 * -2 * 1/t^3 = 4/t^3$
Resulting differential equation:
$y'' + p(t)y' + q(t)y = 0$
$(4/t^3) + p(t)*(-2/t^2) + q(t)*(2/t) = 0$
Simplified to:
$2t^{-2} - p(t)*t^{-1} + 2q(t) = 0$
And then I would assume that there is nothing blatantly wrong with this setup. I think I am missing something though.