Y1=x^4 is a solutionto the ode x^2y"-7xy'+16y=0 use reduction of order to find another independant solution

Answer with explanation:The given differential equation is x²y" -7 x y' +1 6 y=0---------(1)   Let, y'=zy"=z'[tex]\frac{dy}{dx}=z\\\\y=zx[/tex]Substitution the value of y, y' and y" in equation (1)→x²z' -7 x z+16 zx=0→x² z' + 9 zx=0→x (x z'+9 z)=0→x=0 ∧ x z'+9 z=0[tex]x \frac{dz}{dx}+9 z=0\\\\\frac{dz}{z}=-9 \frac{dx}{x}\\\\ \text{Integrating both sides}\\\\ \log z=-9 \log x+\log K\\\\ \log z+\log x^9=\log K\\\\\log zx^9=\log K\\\\K=zx^9\\\\K=y'x^9\\\\K x^{-9}d x=dy\\\\\text{Integrating both sides}\\\\y=\frac{-K}{8x^8}+m[/tex]is another independent solution.where m and K are constant of integration.