In this paper we deal with the problem: when does a differential equation have an elementary solution, that is a solution which can be expressed in terms of algebraic operations, logarithms and exponentials? As an application of our theorem, we give necessary and sufficient conditions for a certain class of first order differential equations to have elementary solutions. For the simplest differential equation yf — a, where a is an algebraic function, Liouville showed that if such an equation has an elementary solution, then this solution is an algebraic function plus a sum of constant multiples of logarithms of algebraic functions. In his paper, Liouville's Theorem on Functions with Elementary Integrals, Pacific J. of Math., 24 No. 1, Rosenlicht showed how this theorem can be handled algebraically and generalized. We will use Rosenlicht's methods to show that if an arbitrary algebraic differential equation has an elementary solution, this solution must be of a special form. An (ordinary) differential field is a field K and a map ': K-* K called a derivation, which satisfies (a + by = a' + V and (ab)r = a'b + abf for all α, b in K. For example, a field of functions, meromorphic in some region of the plane, with the usual differentiation, is such a field. A differential subfield k of K is a subfield which is closed under the derivation. If c is in K and c' = 0 then c is called a constant of K. The set of constants can be seen to form a subfield of K. In this paper all fields will be of characteristic 0. By a differential equation of order n over fc, we mean an expression of the form f(y, y , y{n)) = 0 where / is a polynomial, with coefficients in k, in the variables y,y', , y{n) with y{%) actually appearing. An element of K is said to satisfy such an equation if f(u, u'f , u{%)) = 0 where u{i) is the ΐth derivative of u. We note that if satisfies a differential equation of order n, then the field k(u, u , u{n)) is a differential field of transcendence degree at most n. To see that it is closed under the derivation, note that by differentiating the equation f(u, u , u{n)) = 0 we can solve for u{n+l) in terms of lower derivatives of u. If k c K are differential fields an element u(u =£0) in K is called an elementary integral (exponential of an elementary integral) with respect to k if there exist elements v0, vu ,