Construct an example of a first-order differential equation on $\mathbb{R}$ for which there are no solutions to any initial value problem.
Could anyone please get me started on this. I am struck as to which direction to go
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You want non-existence of solutions locally or globally; the latter is easy to give examples (I gave a very simple one here); the former, I can't think of anything (due to Carathéodory's existence theorem, for example) – Mariano Suárez-Álvarez Feb 08 '14 at 05:51
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1$(y')^2+1=0$. ${}{}$ – André Nicolas Feb 08 '14 at 06:05
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@ André Nicolas: could u explain your answer a little? – afsdf dfsaf Feb 08 '14 at 06:09
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1I take it we are talking about functions from the reals to the reals. Since $(y')^2+1\ge 0$ always, there is no (real) solution to the DE $(y')^2+1=0$. Same for $(y')^2+x^2+1=0$. – André Nicolas Feb 08 '14 at 06:17
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Simply consider $\dot x=f(t)$ where $f$ is the Dirichlet function.
There's no solution for any initial value since $f$ is discontinuous everywhere, while the derivative of a differentiable function should be continuous somewhere.
It's a direct consequence of Baire category theorem, since for any differentiable function $g$, $g'(t)=\lim_{n\to\infty}n(g(t+1/n)-g(t))$, a pointwise limit of a sequence of continuous functions.
For more details, and the continuity of a derivative, see https://math.stackexchange.com/a/112133/23875
