Consider the differential equation $$y''=-c/y^2,$$ describing the motion of a ball you throw straight up in the air, high enough that you can't assume the force of gravity is constant.
We certainly can't solve it by any method we cover in class, nor by any method I know. I like to show how we can derive conservation of energy and hence escape velocity from the DE even though we can't solve it. But
Question: Can anyone state definitively that there is in fact no closed-form solution? (For general initial conditions...)
Edit: Thanks to various comments, I realize I was being a little dumb - I had conservation of energy, which, as I should have realized years ago, is actually a separable first-order equation, leading at least to implicit solutions: It's easy to see that $$\left(\frac12(y')^2-\frac cy\right)'=0,$$so $$\frac12(y')^2-\frac cy=k,$$which is separable. (You end up with an integral that's basically $$\int\frac{dy}{(1+1/y)^{1/2}},$$with a few irrelevant constants. The idiot-freshman substitution $u=(1+1/y)^{1/2}$ converts that to the integral of a rational function...)
There are implicit solutions $$ t = a + c k^3 \ln\left(c k^2 + y + \sqrt{y^2 + 2 c k^2 y}\right) - k \sqrt{y^2 + 2 c k^2 y}$$ and $$ t = a - c k^3 \ln\left(c k^2 + y + \sqrt{y^2 + 2 c k^2 y}\right) + k \sqrt{y^2 + 2 c k^2 y}$$
But we would not expect these to have explicit closed-form solutions for $y$ as functions of $t$.