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.
Best Answer
Take the initial value problem $$y'=\cases{x\bigl(1+2\log|x|\bigr)\quad &$(x\ne0)$ \cr 0&$(x=0)$\cr}\ ,\qquad y(0)=0\ .$$ This example obviously fulfills the assumptions of the existence and uniqueness theorem, so there is exactly one solution. As is easily checked this solution is given by $$y(x)=\cases{x^2\>\log|x|\quad&$(x\ne0)$\cr 0&$(x=0)$\cr}\ .$$ This function is not analytic in any neighborhood of $x=0$.