Kovacic algorithm for differential equations of the type
where are polynomials in
(implemented in Macsyma)
Singer algorithm for differential equations of the type
where are rational or algebraic functions
(not implemented until now due to its massive complexity)
algorithms that look for Liouville type solutions (i.e.,
rational functions extended by algebraic functions, exponential
functions and integrals) or decide if such a solution exists
are polynomials in 
(implemented in Macsyma)
are rational or algebraic functions
(not implemented until now due to its massive complexity)