In [G. F. Simmons' Differential Equations book (p.141)](http://books.google.com/books?id=JVWPPQAACAAJ&dq=G+F+Simmons+Differential+equations&hl=en&ei=yBZNTc3cGdH-4Abu-amoCQ&sa=X&oi=book_result&ct=result&resnum=2&ved=0CC4Q6AEwAQ), the following claim is made:
“... As a matter of fact, there is **no known type** of second order linear differential equation- apart from those with constant coefficients, and those reducible to these by changes of the independent variable--which can bee solved in terms of elementary functions.”


About 36 years have now passed since this statement made its published appearance. Is the remark still true or was it false even before?


My motivation is the analogous theory of integrals of finite combinations of elementary functions where you know certain large and important classes of functions whose integral is expressible as an elementary function( or as a finite combination of such). As we know, there is a somewhat complete classification as to which integral of finite combination of elementary function of one variable is expressible as a finite combination of elementary functions.(References are [Piskunov's book](http://rads.stackoverflow.com/amzn/click/0677206003) and [this](http://www.solitaryroad.com/c351.html) ). So it seemed to me that it is natural to care whether such theorems existed for differential equations.


Thanks.



Is this 1974 claim still  valid?
This claim is *not* valid. The big breakthrough was the paper below:


[MR0839134](http://www.ams.org/mathscinet-getitem?mr=839134) (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.


