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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:

  1. How did that become evident?

  2. What remarkable changes has this point of view brought about?

Thanks.

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It would be helpful to hear about the context (in particular: about the author) of this paragraph. "Few things in the history of mathematics" have a different meaning for different people. –  Christian Blatter Sep 13 '12 at 18:00
    
I found it here –  Shahab Sep 13 '12 at 18:13

3 Answers 3

up vote 12 down vote accepted
+50

Your question would deserve a very long and complicated answer. To summarize, you may think of the development of the concept of function. At the very beginning, a function was an equation, i.e. a formula $y=\ldots$ written by elementary "atoms" (powers, logarithms, sines, cosines, etc). It then become clear that an "abstract" idea of function was more useful than that: sometimes there is no finite combination of "atoms" to write down a function, and yet you can study its properties, like $\int e^{x^2}\, dx$.

In the realm of differential equations, most equations do not have explicit, elementary, solutions. Solutions are found by integrating complicated expressions, sometimes they are found implicitly, and there is no hope to write a solution as $y=\ldots$

I am not an expert of history, but the breakthrough happened when functional analysis began to develop. A differential equation is simply (!) and equation whose unknown lies in a function space. From this viewpoint, it is more important to study qualitative properties of the solutions, rathen than to write down a crazy formula with power series, special functions, and so on. I believe that the study of differential equations was a stimulating problem for a large part of modern mathematical analysis, in the last 150 years (more ore less). Just think of nonlinear functional analysis, that was born essentially to solve differential equations as fixed-point or variational problems.

I cannot say if there was a precise reason, in the hstory of mathematics, why mathematicians had to abandon explicit solutions. The necessity probably grew up slowly.

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A simple reason one can't hope for exact "closed form" solutions is there are differential equations that can't be "solved", i.e. it's known they have solutions that do not have a "closed form", the solutions are inexpressible in terms of any functions that exist in the literature.

On the more extreme end, you can cook-up differential equations where the properties of the solutions are known to be incomputable. This is much like the fact that there are real numbers that are incomputable -- in principle it is impossible to write a computer algorithm to crank-out the decimal expansion of these numbers. Basically this is because the number of computer algorithms one can write is countable, but the number of real numbers is not.

There's a fairly long MO thread on this topic that supplies lots of additional detail:

http://mathoverflow.net/questions/15292/why-cant-there-be-a-general-theory-of-nonlinear-pde

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I think the reason is the qualitative and chaotic behaviour of the n-body problem. I believe chaos and n-body problems were the cause of the change.

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