The existence and uniqueness is usually stated when the coefficient of \(\displaystyle{y}{''}\) is 1, so you want to look at

\(\displaystyle{y}{''}{\left({x}\right)}-{\frac{{{4}}}{{{x}}}}{y}'{\left({x}\right)}+{\left({1}+{\frac{{{6}}}{{{x}^{{2}}}}}\right)}{y}{\left({x}\right)}={0}\)

There is no contradiction because the coefficients

\(\displaystyle-{\frac{{{4}}}{{{x}}}}\) and \(\displaystyle{1}+{\frac{{{6}}}{{{x}^{{2}}}}}\)

are continuous except at x=0

\(\displaystyle{y}{''}{\left({x}\right)}-{\frac{{{4}}}{{{x}}}}{y}'{\left({x}\right)}+{\left({1}+{\frac{{{6}}}{{{x}^{{2}}}}}\right)}{y}{\left({x}\right)}={0}\)

There is no contradiction because the coefficients

\(\displaystyle-{\frac{{{4}}}{{{x}}}}\) and \(\displaystyle{1}+{\frac{{{6}}}{{{x}^{{2}}}}}\)

are continuous except at x=0