Many years ago, I considered the following non-linear differential equation:
$y=y''\cdot(1+y'^{2})^{-3/2}$


This equation expresses the equality between the value of a given function $y\in C^{2}(R)$ and the value of its curvature at the same point.


I almost convinced myself that there is no nontrivial solution to this equation, but couldn't find any rigorous argument to prove it.


I'm thus looking for techniques and or references to get a real proof of this expected result.


Many thanks in advance.



Techniques to solve a non-linear differential equation related to curvature
Well the standard techniques would take advantage of the fact that the equation doesn't explicitly involve the independent variable $x$ to integrate the equation once, thereby leading to the conservation law
$$
y^2 + 2(1+y'^2)^{-1/2} = r^2,\tag1
$$
where, $r>0$ is a constant. Note that we must have $|y| < r$


The relation (1) can then be solved to yield, after separation of variables, 
$$
\frac{(r^2-y^2)\,dy}{\bigl((2-r^2+y^2)(2+r^2-y^2)\bigr)^{1/2}} = dx.\tag2
$$
The left hand integral can be computed in terms of elliptic functions, of course, but you don't need to do this to do a qualitative analysis.


If $r>\sqrt2$, then we must have $|y|\ge\sqrt{r^2-2}$, so, replacing $y$ by $-y$, we can assume that $\sqrt{r^2-2}\le y(x) < r$. Because the integral on the left hand side of (2) over the interval $\sqrt{r^2-2}\le y \le r$ is finite, it follows that $y(x)$ can only be defined over an $x$-interval of finite length. Thus, there is no entire solution in this case.


If $r = \sqrt2$, then, again, we could only have $y=0$ where $y'=0$. However, the left-hand integral of (2) over the interval $0<y<a$ when $0<a<\sqrt{2}$ is infinite, so we cannot have $y$ vanishing for any finite value of $x$, thus, $y$ cannot vanish and we can assume $y>0$. The above integral then leads to a solution $y(x)$ on an interval of the form $(-\infty,C]$ where $C<\infty$, so there is no entire solution in this case either.


Finally, if $r<\sqrt{2}$, then we have $|y| < r$, but the integral of the lefthand side of (2) over the interval $-r < y < r$ is finite, so, again, there is no solution defined over the entire $x$-axis.


