How complex can a Differential Equation be?

Question

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 $\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{\mathrm{d}y/\mathrm{d}x}$.

Answer 1

I guess a pretty general ("complex") form is:

$$\dot{y}=Ay$$

With $y(0)=y_0$

Where $A$ is some usually, but not necessarily, linear operator.

This is known as an abstract Cauchy problem. If $A$ is linear, and some fancier words like self adjoint and bounded, then:

$$y(t)=e^{At}y_0$$

This is a pretty general form, and leads to what's called $c_0$ semi group theory.

More complicated differential equations might be something I study, SPDEs. Take your favorite ODE or PDE and add a term $\dot{W}$. $W$ is some noise term called Brownian motion and $\dot{W}$ is the derivative of Brownian motion. However Brownian motion is not differentiable. How's that for complicated? :)

Examples:

$$\partial_t u=\partial_{xx} u +u\dot{W} \text{ (Stochastic Linear Heat)}$$

$$\partial_t u=\partial_{xx} u +(\partial_x u)^2-\infty+\dot{W} \text{ (KPZ)}$$

Note that these are special cases of a Cauchy problem. The first one being linear, the second one not.

Answer 2

As requested here is a very succinct way of writing a very broad class of non-local differential equations $$f\left(\underline{x},\left\langle \delta_j(\underline{x},\varphi_{k}(\underline{x})),\varphi_{i}(\underline{x})\right\rangle \right)=0$$ where $\underline{x}$ varies over some arbitrary manifold $M$ (if you wanted to go really crazy I guess it could be an arbitrary diffeology, but I don't think anybody uses those), $\varphi_i(\underline{x})$ are a family of real valued functions (technically they could take on values in an arbitrary manifold, but I didn't want to figure out how the next part generalizes. Although it's not 100% necessary because you can talk about coordinate patches), and $\delta_j(\underline{x},\varphi_k(\underline{x}))$ is an arbitrary family of distributions on $M$ parameterized by $\underline{x}$ and all of the $\varphi_i(\underline{x})$, and $f$ is some arbitrary function of all of those (assuming nothing about its smoothness, and guaranteeing nothing about the existence of solutions).

Although as written this is actually so general that it includes not only all local and non-local differential equations (including fractional derivative differential equations, which can be thought of as delay differential equations), but pretty much any equation between functions, including integral equations (which covers weak solutions of PDEs), polynomials, functional equations, recursion relations on sequences, etc. I don't know if it covers stochastic differential equations (outside of the linear case, I think I remember seeing that linear SDEs can be converted to certain PDEs). It also doesn't cover stuff like 'p-adic differential equations' which use an entirely different field than $\mathbb{R}$. But ultimately you can push to more and more general formalizations but you lose any kind of ability to prove general statements.