Verifying that a solution satisifes the ODE is a matter of substituting the solution into the ODE. For example,
[tex]y_1=x\implies{y_1}'=1\implies{y_1}''={y_1}'''=0[/tex]
[tex]x^3{y_1}'''+10x^2{y_1}''+16x{y_1}'-16{y_1}=16x-16x=0[/tex]
The Wronskian for these solutions is
[tex]W(x,x^{-4},x^{-4}\ln x)=\begin{vmatrix}x&x^{-4}&x^{-4}\ln x\\1&-4x^{-5}&x^{-5}(1-4\ln x)\\0&20x^{-6}&-x^{-6}(9-20\ln x)\end{vmatrix}=25x^{-10}[/tex]
which is non-zero for all [tex]x\in(0,\infty)[/tex], so the solutions are indeed linearly independent.
The general solution is then
[tex]y=C_1x+C_2x^{-4}+C_3x^{-4}\ln x[/tex]
