Chapter 8.5, Problem 19E

To determine

(a)

To show:

For $L\left[y\right]\left(x\right):=x^3{y}^{‴}\left(x\right)+x{y}^{\prime }\left(x\right)-y\left(x\right)$,

$L\left[x^r\right]\left(x\right)={\left(r-1\right)}^3{x}^r$.

To determine

(b)

To show:

The $L\left[y\right]=0$ has the general solution $y\left(x\right)=c_{1}x+c_{2}x\ln x+c_{3}x{\left(\ln x\right)}^2$.