Non-uniqueness for a differential equation and a proof by ChatGPT
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918486070001664 |
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| author | Street, Brian |
| author_facet | Street, Brian |
| contents | Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[
\frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a large class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro, and we include a discussion of its output as well as a complete proof. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04810 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-uniqueness for a differential equation and a proof by ChatGPT Street, Brian Analysis of PDEs Classical Analysis and ODEs 2020: 34A55 (Primary), 34A12, 35A02, 44A10 (Secondary) Let $f(t,x),M(t,x)\in C([0,1]^2)$ with $M(t,x)>0$. We consider differential equations of the form \[ \frac{\partial f}{\partial t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},\quad x>0. \] For a fixed positive weight $M$, we ask whether the condition $f(0,x)=0$ forces $f\equiv 0$. We show the answer is negative for smooth functions: there exist $f(t,x),M(t,x)\in C^{\infty}([0,1]^2)$ with $f(0,x)=0$, $f(t,0)\not\equiv 0$, and $M(t,x)>0$ satisfying the above equation. However, we show that for a large class of $M(t,x)$, the equation does have uniqueness. We relate this to uniqueness/non-uniqueness theorems for weighted Laplace transforms. A key example originated in an output by ChatGPT-5.5-Pro, and we include a discussion of its output as well as a complete proof. |
| title | Non-uniqueness for a differential equation and a proof by ChatGPT |
| topic | Analysis of PDEs Classical Analysis and ODEs 2020: 34A55 (Primary), 34A12, 35A02, 44A10 (Secondary) |
| url | https://arxiv.org/abs/2605.04810 |