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Given the differential equation x^{{3}}y''+4x^{{2}}y'+3y=0\,

i.e.,   y''+{\frac  {4x^{{2}}}{x^{{3}}}}y'+{\frac  {3}{x^{{3}}}}y=0,

y''+{\frac  {4}{x}}y'+{\frac  {3}{x^{{3}}}}y=0,

So P(x)={\frac  {4}{x}},Q(x)={\frac  {3}{x^{{3}}}},

\therefore x=0 is a singular point because the functions P(x) and Q(x) are not analytic at x=0.

To check for regularity consider the functions (x-0),P(x) and (x-0)^{{2}},Q(x)

i.e., x{\frac  {4}{x}} and x^{{2}}{\frac  {3}{x^{{3}}}}

i.e., 4 and {\frac  {3}{x}} which are not analytic at x=0

Hence x=0 is and irregualr singular point