# Simple proof that non-linear DE's do no not satisfy superposition?

I'm wondering if there's a simple proof that solutions to non-linear differential equations do not satisfy the superposition principle?

Some explicit examples would also be great.

Cheers!

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It depends what your definition of linear is in this case. I think it wouldn't be unreasonable to say "a DE is linear if linear combinations of solutions are solutions" which is pretty much "a DE is linear if it satisfies superposition" –  Ben Millwood May 24 '12 at 13:52
For such cases the simplest example that comes to mind usually works. Consider d.e. $y'=y^2$. It has solutions $y_c(x)=\frac 1{c-x}\,$ for $c\in \mathbb R$. But functions $y_1+y_2$ and $2y_1$ are not solutions.
But it cannot be said that all solutions of any nonlinear equation do not satisfy the superposition principle. Consider, for example, equation $(y'')^2=0$.