Intereting Posts

What is an intuitive meaning of $E(\overline { X } )$ and $Var(\overline { X } )$?
Solving for $a$ in this equation $\sin^a(a) = b$
Prove that $m$ is ergodic.
Why $a^n – b^n$ is divisible by $a-b$?
Link between a Dense subset and a Continuous mapping
A non-negative matrix has a non-negative inverse. What other properties does it have?
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$
Simplifying $\sqrt{4+2\sqrt{3}}+\sqrt{4-2\sqrt{3}}$
Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$
Is there a chain rule for integration?
Uncountable $\sigma$-algebra
How do we get the result of the summation $\sum\limits_{k=1}^n k \cdot 2^k$?
Proving inequalities about integral approximation
Integral inequality (Cauchy-Schwarz)
Homology of the Klein Bottle

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!

- Solving a second-order linear ODE: $\frac{d^2 y}{dx^2}+(x+1)\cdot \frac{dy}{dx}+5x^2\cdot y=0$
- Can someone show me a proof of the general solution for 2nd order homogenous linear differential equations?
- dropping a particle into a vector field, part 2
- second order ODE via variation of parameters
- When does Initial Value Problems have: no solutions, more than one solution, precisely one solution?
- Help with solving differential Equation using Exact Equation method

- How to solve differential equations of the form $f'(x) = f(x + a)$
- differential equations, diagonalizable matrix
- Cancelling differentials
- Supplemental reference request-Graduate level PDE problems and solutions book
- one-dimensional inverse square laws
- To what extent can you manipulate differentials like dy and dt like actual values?
- Numerical Analysis References
- Express in terms of Legendre polynomials
- solution of a ODE with a funtion of $\dot{x}$
- Exponential of matrices and bounded operators

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 $x=0$.

- Figure out if a fourth point resides within an angle created by three other points
- How prove this $\frac{a}{(bc-a^2)^2}+\frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2}=0$
- Is the orbit space of a Hausdorff space by a compact Hausdorff group Hausdorff?
- Prove that it is impossible to have integers $b=5a$ under a digit re-ordering
- Strategies For Summing Harmonic Numbers
- Finding all solutions of $x^{11}\equiv 1\bmod23,$
- primitive element $a$ of $\mathbb F_{p^n}/\mathbb F_p$ such that $a^n\in\mathbb F_p$
- Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$
- A question on ring homomorphisms and maximal ideals.
- Factorizing polynomial $x^5+x+1$
- Linear isoparametrics with dual finite elements
- What are the applications of finite calculus
- Count number of exact matching sequences
- Geometric interpretation of the multiplication of complex numbers?
- Calculate limit with summation index in formula