  The purpose of this paper is to study a class of ill-posed differential
equations. In some settings, these differential equations exhibit uniqueness
but not existence, while in others they exhibit existence but not uniqueness.
An example of such a differential equation is, for a polynomial $P$ and
continuous functions $f(t,x):[0,1]\times [0,1]\rightarrow \mathbb{R}$,
\begin{equation*} \frac{\partial}{\partial t} f(t,x) = \frac{
P(f(t,x))-P(f(t,0))}{x}, \quad x>0. \end{equation*} These differential
equations are related to inverse problems.
