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I would like an example of a ODE for which there is no solution. Online I could only find this post which however would imply non existence of a $C^1$ solution, not an absolutely continuous one. So I am looking for a function $g(t,x)$ such that

$$ \begin{cases} y' = g(t,y) \\ y(0)=y_0 \end{cases}$$

does not admit a solution in $AC(I;\mathbb{R})$, with $I$ being a neighborhood of 0.

tommy1996q
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  • See Carathéodory's existence theorem perhaps https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem – Gribouillis Jan 30 '25 at 10:23
  • Yeah I know it, but there it only gives an example of non existence of a $C^1$ solution – tommy1996q Jan 30 '25 at 10:27
  • The other examples were for simple ``search of a primitive'' problems. An example for a $g$ discontinuous in $y$: consider $g(t,y)=-\text{sgn}(y)$ with $sgn(0)=1$ and initial condition $y(0)=0$. – MatteoDR Jan 30 '25 at 13:04

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There are differentiable function which are not absolutely continuous. See non AC. Take any such function $y_0$ and then consider the differential equation $y’=g(t)$ where $g$ is the derivative of $y_0$.

Alternatively, the primitive of a Lebesgue integrable function is absolutely continuous, so take $g(t)$ to be the characteristic function of a non Lebesgue measurable set.

Gio67
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