This claim is *not* valid. The big breakthrough was the paper below:

MR0839134 (88c:12011) 
Kovacic, Jerald J.
An algorithm for solving second order linear homogeneous differential equations. 
J. Symbolic Comput. 2 (1986), no. 1, 3–43. 
12H05 (34A30) 

Later, M. Petkovsek extended the ideas to second order *difference* equations.