15
$\begingroup$

In many talks, I have heard people say that the differential equation they are interested in has no analytical solution. Do they really mean that? That is:

Can you prove a differential equation has no analytical solution?

I suspect what they mean is that no one has been able to derive one, but I could be wrong. I also have a question related to the former case.

What are some simple examples of differential equations with no known analytical solution?

The differential equations courses at my university are method based (identify the DE and use the method provided) which is completely fine. However, I'd like to have some examples which look easy (or look similar to ones for which the given methods will work) in order to show students that not all differential equations are so easily solved.


Added later: Taking the comments into account, I suppose the type of differential equations I am looking for in the second question are ones which, at this point in time, can only be solved using numerical methods (which, as Emmad Kareem points out, would be good motivation for learning such methods).


The kind of thing I'm looking for: I was talking to my friend who does Fluid Mechanics and he suggested the Blasius equation $$f''' + \frac{1}{2}ff'' = 0.$$ Apart from $f(x) = ax + b$, there are no known (as far as he knows) analytical solutions.

  • 4
    It very much depends on what you mean by [closed-form/analytical solutions](http://math.stackexchange.com/questions/9199/what-does-closed-form-solution-usually-mean). For example, for suitably restrictive definitions of closed form, [Bessel's differential equations do not admit closed form solutions](http://en.wikipedia.org/wiki/Bessel_function), but most ODE users would consider Bessel functions well known and perhaps part of their "elementary" toolkit.2012-10-10
  • 2
    The cynical side of me wants to say: if you can't express the solution of a "simple looking" ODE in analytical terms (allowing the use of special functions), then you might as well define a new special function that refers to the solution of that ODE....2012-10-10
  • 0
    @WillieWong: I did consider your first point and I suppose it depends on what you mean by a *special function*. I know that differential Galois theory comes into the picture here as there is some problem with trying to capture all of the special functions. As J.M.'s answer to the question you linked to indicates, there isn't a universally accepted definition for 'closed form', but I would consider Bessel's functions to be analytical solutions. I'm not sure whether I can change my second question so that I can avoid this issue.2012-10-10
  • 2
    You'd imagine that every book on numerical methods (and its directly related subjects) should start with an answer to your question. However, this is not the case.2012-10-10

6 Answers 6