Abstract: Let C(x) be the rational function field over the complex numbers equipped with the usual derivation. Let f be an irreducible polynomial over C(x) in n+1 variables. Let Y be a differential indeterminate. Substitute Y and each of its derivatives up to order n into f. The result is called an algebraic differential equation (ADE) over C(x) of order n. In this talk, we will provide a necessary and sufficient condition for ADEs to admit infinitely many solutions that are differentially algebraically independent over C(x). We revisit certain classical results of Goldman, Singer and Rosenlicht on solutions of first order nonlinear differential equations and explain how our results can be used to extend these results to higher order algebraic differential equations. Our approach uses the Galois theory of linear differential equations, which we will briefly introduce in the talk. This is a joint work with Partha Kumbhakar and Ursashi Ro.