Solvability v.s. the existance of a solution for nonlinear ODE's

[pergradus]

In physics we're told often that there is no analytical way to solve most nonlinear differential equations (I don't know if this can be proven or not, or it's just assumed because no one has found a way to do it), so we use a computer to solve them numerically.

I'm wondering though, assuming it is impossible to solve a differential equation by analytical means, does that mean that there is no analytical solution which exists? That is, could you somehow guess some well-defined function that satisfies the nonlinear equation, even if there's no way of directly solving the equation for it?

 

[Number Nine]

In many cases (most, actually) there just doesn't exist a solution in closed form. The functions you're used to (i.e. functions that have closed form representations) are only a very small subset of all the possible functions; it's natural that they would satisfy only a small subset of PDE's and ODE's.