Differential Equations without Analytical Solutions 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.
 A: Yes, it can be shown that differential equations do not have analytic solutions, using differential galois theory. An example is the second order linear ode
$y''+ x y' = 0$
this should be a good example for your purposes as it is only slightly different from the second order linear odes with constant coefficients which are easily solved.
PS: I assume you understand the difference between the ode not having an analytical solution and it not having a solution at all.
A: Hint: I have posted   an  example here in MO for a differential Equation which has no analytical solution that is :$\displaystyle \ f'={e}^{{f}^{-1}}$ . with nice  answer for Non -existence of analytic solution of it  by Christian Remling
A: 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 $x=0$.
A: There's something worse than having no analytical solution. Pour-El and Richards found an ordinary differential equation $\phi'(t)=F(t,\phi(t))$ with $F$ computable and no computable solution. A reference is Marian Boykan Pour-El and Ian Richards, A computable ordinary differential equation which possesses no computable solution, Ann. Math. Logic 17 (1979), no. 1-2, 61–90, MR0552416 (81k:03064). 
A: For an example, why not start with the first differential equation ever, Newton's DE for falling objects, derived in high school physics class, you remember, $$A = \frac{G M_{earth}}{R^{2}}.$$  It's the universities' little secret that most differential equations describing the real (non-linear) world are not solvable. So, they don't tell you in high school that the equation is not solvable.  They don't tell you in your university physics class (I've checked the text used by my local U), or even the classical mechanics class (this text notes that a modified equation is solvable and can be 'inverted' to get a solution, but they don't mention that the 'inversion' is not closed, i.e. is a procedure for approximation).
A: In introductory physics, the equation of motion for a pendulum is given as $$ml^2\frac{d^2\theta}{dt^2}=-mgl\theta$$. This gives you an easy solution in sines and cosines. That uses the "small angle approximation". The true equation is 
$$ml^2\frac{d^2\theta}{dt^2}=-mgl\sin\theta$$, admitting no "analytical/closed form solutions", depending on how you define that. 
