The following table gives an example ode for each case of the above. Recall there are 4 cases. Case 1 \(\left ( n=1\right ) ,\) and case 2 \(\left ( n=2\right ) \) and case 3 \(\left ( n=4,6,12\right ) \) and case 4 which means no Liouvillian solution exist. Recall also that if ode belong to case 1,2 or 3, this does not imply that Liouvillian solution exists. For example, below \(x^{2}y^{\prime \prime }-2xy^{\prime }+\left ( x^{2}+2x+2\right ) y=0\) satisfies conditions for case \(1\), however, we can find out that no Liouvillian solution exists.
| case number | ODE |
| One \(L=\left [ 1\right ] \) | \(x^{2}y^{\prime \prime }+4xy^{\prime }+\left ( x^{2}+2\right ) y=0\) |
| Two \(L=\left [ 2\right ] \) | \(2xy^{\prime \prime }-y^{\prime }+2y=0\) |
| One and Two \(L=\left [ 1,2\right ] \) | \(4x^{2}y^{\prime \prime }+4x\left ( 1-x\right ) y^{\prime }+\left ( 2x-9\right ) y=0\) |
| One \(L=\left [ 1\right ] \) | \(x^{2}y^{\prime \prime }-2xy^{\prime }+\left ( x^{2}+2x+2\right ) y=0\) |
| One and Two and Three \(L=\left [ 1,2,4,6,12\right ] \) | \(y^{\prime \prime }-\frac {1}{\left ( 4x^{2}\right ) }y=0\) |
| Case 4, (i.e. No Liouvillian solution exist) | \(y^{\prime \prime }-xy=0\) |