### 2 Examples of ODE’s for each case

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) $$y^{\prime \prime }-xy=0$$