Non-uniqueness for a differential equation and a proof by ChatGPT

Let $f(t,x),M(t,x)\in C([0,1{]}^2)$ with $M(t,x)>0$. We consider differential equations of the form $$\frac{\mathrm{\partial }f}{\mathrm{\partial }t}(t,x)=\frac{M(t,x)f(t,x)-M(t,0)f(t,0)}{x},x>0.$$ 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 $f(t,x),M(t,x)\in C^{\mathrm{\infty }}([0,1{]}^2)$ with $f(0,x)=0$, $f(t,0)\equiv ̸0$, 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.