The differential equation $$x^{2}\frac{d^2y}{dx^2}-7x\frac{dy}{dx}+16y = 0$$ has $x^{4}$ as a solution.
Applying reduction order we set $y_2 = u x^{4}$.
Then (using the prime notation for the derivatives)
$y_2^\prime =$
$y_2^{\prime\prime} =$
So, plugging $y_2$ into the left side of the differential equation, and reducing, we get


The reduced form has a common factor of $x^5$ which we can divide out of the equation so that we have $x u^{\prime\prime}+ u^\prime=0$.
Since this equation does not have any u terms in it we can make the substitution $w = u^\prime$ giving us the first order linear equation $xw^\prime+w = 0$.
This equation has integrating factor for x > 0.
If we use a as the constant of integration, the solution to this equation is $w =$
Integrating to get u, and using b as our second constant of integration we have $u =$
Finally $y_2 =$ and the general solution is