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

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.

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.

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

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 trivial 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.

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

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.