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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!

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    See the [Picard-Lindelöf theorem](http://en.wikipedia.org/wiki/Picard_existence_theorem).2010-11-02

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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).

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    An interesting detail about Lew$y$'s paper is that it doesn't cite an$y$ other works.2010-11-02
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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.

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    They are known. See, e.g. www.deaneyang.com/papers/gunther.pdf2010-11-02
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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 $x=0$.

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    @Matt: The solution is a piecewise quadratic function. It is $-x^2/2 + C$ for $x \leq 0$ and $+x^2/2 + C$ for $x \geq 0$. I got this by integrating $|t|$ from $0$ to $x$.2010-11-02