Another Cauchy-Euler ODE. Take [tex]y=x^r[/tex] and you get
[tex]r(r-1)x^r-16rx^r+64x^r=0[/tex]
[tex]r^2-17r+64=0[/tex]
[tex]\implies r=\dfrac{17}2\pm\dfrac{\sqrt{33}}2[/tex]
so that the general solution is
[tex]y=C_1x^{(17+\sqrt{33})/2}+C_2x^{(17-\sqrt{33})/2}[/tex]
With the given boundary conditions, you have
[tex]y(1)=0\implies 0=C_1+C_2[/tex]
[tex]y(2)=1\implies 1=C_12^{(17+\sqrt{33})/2}+C_22^{(17-\sqrt{33})/2}[/tex]
[tex]\implies C_1=\dfrac{2^{(-17+\sqrt{33})/2}}{2^{\sqrt{33}}-1},C_2=-\dfrac{2^{(-17+\sqrt{33})/2}}{2^{\sqrt{33}}-1}[/tex]
