Summary: It is proved that the differential equation \[ x^ 2 f''(x)+ (\beta_ 0 x^ 2+ \beta_ 1 x) f'(x)+ (\gamma_ 0 x^ 2+ \gamma_ 1 x+ \gamma_ 2) f(x)=0 \] for arbitrary \(\beta_ 0\), \(\gamma_ 0\), \(\gamma_ 1\) and \(\beta_ 1\neq 0,\pm 1,\pm2,-3, -4,\dots\) has an entire solution if and only if the condition \(\beta_ 1-2- \gamma_ 2= n^ 2+ (\beta_ 1- 3)n\), \(n=1,2,\dots\) is satisfied. The entire solution \(f(x)\) for \(n=k\), \(k\geq 1\) is of the form \(f(x)= x^{k-1} \sum_{k=1}^ \infty \alpha_ k x^{k-1}\), \(\alpha_ 1=1\).