This seems to be caused by the use of `\label{}`. Here is your first equation, with and without that command:

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} \tag{1}\label{1}$$

$$ h(t)\nabla \times \tilde {\mathbf H} = \mathbf J +\epsilon_0 \dot {e}(t) \tilde {\mathbf E} \tag{1}$$

Both look identical during editing, but the first is sometimes (but inconsistently) mangled in the output.

I'm unaware of how `\label{}` works in MathJax, and whether it is truly supported. But do you really need it? Why not use just `\tag{}`s, and hard-code any references in the text? (i.e. use `$(1)$` instead of `$\eqref{1}$`.)