5.1.2.2 Example 2

\[ r=x \]

There is one pole of order zero (an even pole). So case 1 or 3 qualify. But \(O\left ( \infty \right ) =0-1=-1\) which is odd. But case 1 and 3 require \(O\left ( \infty \right ) \) be even. Hence case 1,2,3 all fail. This is case 4 where there is no Liouvillian solution. This is known already, because this is the known Airy ode \(y^{\prime \prime }=xy\). Its solution are the Airy special functions. These are not Liouvillian solutions. Hence \(L=\left [ {}\right ] \)