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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.


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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
up vote 4 down vote accepted

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 $(y'')^2=0$.

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Is there some general proof (barring trivial solutions), or am I looking for something that can't be found? – trolle3000 May 24 '12 at 17:36
@trolle3000 Proof of that exactly? An example is given in the answer. Also given an example of a non-linear equation all solutions of which satisfy the superposition principle. – Andrew May 24 '12 at 17:44

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