# What is an example of a second order differential equation for which it is known that there are no smooth solutions?

I would really appreciate if someone could just write down for me one example of a second order, or higher, differential equation for which it is known that there are no smooth solutions; and it’s fine if it’s a partial differential equation.

At first I thought it would be easy to either come up with an example or else find one by searching google/wiki/arxiv; but now I am not so sure.

I have a thing for non-smooth functions, and it just bothers me that I don’t even know a single example of this type of differential equation. Thanks!

#### Solutions Collecting From Web of "What is an example of a second order differential equation for which it is known that there are no smooth solutions?"

There are already first order linear partial differential equations with smooth coefficients which do not admit any smooth solutions.

Hans Lewy produced the first example of such a PDE. The equation reads
$$\left[-i\partial_x+\partial_y-2(x+iy)\partial_z\right]u(x,y,z)=f(x,y,z),\qquad(x,y,z)\in\mathbb R^{3}.$$
The equation does not have distribution solutions in any neighbourhood of any point in $\mathbb R^3$ provided $f=f(x,y,z)$ is not a real analytic function (it can be smooth though).

The original paper by Lewy is nice, clear and less than 4 pages long (freely available here).

Consider the partial differential equations associated to the isometric embedding problem of the hyperbolic plane into Euclidean 3-space. In $C^1$ there exists a solution by Nash-Kuiper theorem, but it is known classically that there cannot be solutions that are twice or more continuously differentiable.

How about you take the differential equation

$\frac{dy}{dx} = |x|$

This is a linear non-homogeneous differential equation, whose solution is $C^1$ but not smooth at $x=0$.