# What comes after Differential Equations?

First of all, please do excuse the lack of correct terminology, I've haven't learnt Differential Equations at school (yet) so this question comes from just a bit of research I did for my own enjoyment.

I was reading up on differential equations and the first thing I read was that their result is either a function or a family of functions. So I thought, if the result of functions are numbers and the result of eifferential equations are families of functions, is there anything whose results are a family of differential equations?

Since I don't know the terminology of the subject, I don't know what to search on Google to find the answer so I come to you for help. What comes after differential equations?

Thanks a lot

EDIT: I didn't word the question correctly. Sorry about that, I'll try to give an example.
In this normal equation $x^2+2x-3=0$ the solutions are $x_{1}=-3$ and $x_{2}=1$. The solutions are numbers.

In this differential equation $\frac{dx}{dt} = 5x -3$ the solution is $$x(t) = Ce^{5t}+ \frac{3}{5}.$$

The solution is a function/a normal equation.
(Took the example for the differential equation from this page http://mathinsight.org/ordinary_differential_equation_introduction_examples )

What I want to know is if there is a type of equations whose solutions are differential equations.

• Partial Differential Equations, Stochastic Differential Equations ... – Amzoti Nov 22 '13 at 19:31
• Math isn't linear. There are lots of directions you can go in once you're past the usual calculus sequence (and, at least in theory, you could start these courses before finishing up with calculus). I don't know whether you consider linear algebra to be a pre-requisite for differential equations, but a good knowledge of linear algebra is essential for almost anything you would want to study if you were to continue past that point. – Brett Frankel Nov 22 '13 at 19:32
• Thanks for the answers but it seems I didn't explain my question correctly, I was not asking what I should study after differential equations. Sorry about that. Could you please check my original question again? – Constantine Loukas Nov 23 '13 at 15:06
• I've never seen a problem whose solution is another problem. – Siminore Nov 23 '13 at 15:18
• I agree with Siminore. IMHO, your question is clear but this train of thoughts is not leading you anywhere. While it may well happen that a differential equation spawns another differential or algebraic equation, there is no "hierarchy of equations" as you are trying to figure it. – Giuseppe Negro Nov 26 '13 at 15:55

Mathematics is not a hierarchy of ever more complicated kinds of equations.

Differential equations are important because among other things they have provided a language for physics to discuss many problems and understand the behaviour of their systems. From that point of view, differential equations are nothing but numerical equations that hold at many points.

But mathematics is way way way more than that. It about logical structures (loosely motivated by number systems) and their relations.

To answer your question more specifically as suggested by nayrb, I don't think there is some standard kind of frame where one writes equations of differential equations. Note that the term "equation" implies that you have some object that you do operations with: in your numeric and differential equations, you can add and multiply numbers and functions respectively. To write equations of differential equations, you should define operations between differential equations. I'm not saying it is not possible, but it is hard for me to imagine how to do it.

• This is true, but the original poster still asks a fair question. – abnry Nov 26 '13 at 15:51
• Fair point. I've added a paragraph. – Martin Argerami Nov 26 '13 at 15:55

Think about doing Calculus I,II,III and DFQ as sitting on a catapult going through the preparation to be launched. Once you have completed DFQ, you are being shot into the sky. And as you know, sky is the limit. So as mentioned by others, you have formed a basis of math knowledge that allows you to embark on different paths, ranging from real analysis, complex analysis, matrix algebra, diff geometry etc.

• Hi! Thanks for the answer. I updated my question because I didn't manage to explain what I was wondering correctly, would you mind checking it again? – Constantine Loukas Nov 23 '13 at 15:07

I completely agree with Martin's answer, and I believe it is the correct one. Nevertheless, I would like to attempt to more directly answer the question as posed.

There are certain situations where a differential equation is the solution of a problem. One simple example would be to ask for a differential equation which satisfies certain properties. I did ask a question like that, here, and obtained some great answers.

Partial differential equations play a very important role in physics, and many problems in modeling of physical systems amounts to correctly figuring out how to set up a system of partial differential equations. I will note that Newton's discovery of the laws of motion is another example of a "problem" whose "solution" is a differential equation.

In pure mathematics these types of situations are much rarer, but I can think of one. Partial differential equations play an important role in the theory of surfaces, and also manifolds. There is a problem which asks for an isometric imbedding of a manifold as surface in $\mathbb{R}^N$, for which the solution is a system of partial differential equations, See here.

I think the answers given here are so busy trying to to tell you how to think about mathematics that they didn't bother to tell you the answer to your question. There is nothing wrong with how you are thinking. After ordinary differential equations there is the calculus of variations. In this subject, you learn (loosely) that to find an extrema of a functional you can take the euler-lagrange operator on the functional and this will yield a set of differential equations. You can then solve these diff eqs to find the equations of motion of your system.