When I was first being introduced to differential equations, my teacher began by asking the other student and me to write down the **most complex differential equation we could think of** and gave us 10 or 20 seconds to do so, before judging what we'd written and choosing a winner. The purpose of this was mainly to check we knew was a differential equation *was* and to make us think about what they might look like.

My question is, what is the most complex differential equation that could reasonably be written down in 10 seconds? The only restriction is that it must be a DE - use your imagination!

This requires some kind of definition of the word 'complex' in this setting. To illustrate what I mean, I will provide two examples which I would consider to be no more complex than each other.

$$\begin{align}
\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+\frac{\mathrm{d}y}{\mathrm{d}x}=\sin(x+y)\\
\frac{\mathrm{d}}{\mathrm{d}x}\left(5\log x\frac{\mathrm{d}y}{\mathrm{d}x}\right)=\exp(-\sin(7x^3\sqrt y)
\end{align}$$

These are both second order and can be written as $y''+f(x)y'=g(x, y)$. So I suppose what I am looking for is the most complicated **form** of differential equation that you can come up with.

**EDIT:**

Don't let my example mislead you into thinking there are restrictions. I would recommend involving complex numbers, integrals, vectors, perhaps matrices, perhaps something like $\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{\mathrm{d}y/\mathrm{d}x}$.