As I used to understand the primary aim of a student learning differential equations is that given a differential equation he should be able to solve it. However while recently reading a note on the history of differential equations I came across the following paragraph:
A new era began with the foundation of what is now called function theory by Cauchy, Riemann, and Weierstrass. The study and classification of functions according to their essential properties, as distinguished from the accidents of their analytical forms, soon led to a complete revolution in the theory of differential equations. It became evident that the real question raised by a differential equation is not whether a solution, assumed to exist, can be expressed by means of known functions, or integrals of known functions, but in the first place whether a given differential equation does really suffice for the definition of a function of the independent variable (or variables), and, if so, what are the characteristic properties of the function thus defined. Few things in the history of mathematics are more remarkable than the developments to which this change of view has given rise.
My questions are:
How did that become evident?
What remarkable changes has this point of view brought about?