Attempt $1$:$20$ attempts remaining.

The differential equation

$x^2\frac{d^{2}y}{d{x}^2}-7x\frac{dy}{dx}16y=0$

has $x^4$ as a solution.

Applying reduction order we set $y_2=u{x}^4.$

Then $($using the prime notation for the derivatives$)$

$y_2^{\text{'}}=$

$y_2^{\text{'}\text{'}}=$

So$,$ plugging $y_2$ into the left side of the differential equation, and reducing, we get

$x^2{y}_2^{\text{'}\text{'}}-7x{y}_2^{\text{'}}16y_2=$

$$