From differentials to differential equations Suppose I have a function of time $t$ and position $(x,y)$ such that
\begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + p_y \,dy\end{equation}
where the subscript denotes a differentiation. In this case, I am able to derive a   (partial) differential equation from this form.
I'd love to have your help to address the case in which, for example, $dy$ appears also with higher orders. Something like:
\begin{equation} p_t \,dt = p \,dy - p_x (1-x) \,dx + (dy)^2 (p-(1-y)p_y). \end{equation}
or simpler (the key point is the presence of $(dy)^2$). I expect that in this case the pde will be second order...
Any idea?
P.S. I posted yesterday a similar question on the math.stackexchange but maybe it is more a physics-like question :)
 A: The $(dy)^2$ term is totally negligible, it's as if it was not there. If you had two differentials everywhere, then yes, it would lead to a 2nd order diff. equation.
A: I am taking the comments and equations in the OP edits as part of this question which therefore has both a mathematical part and a physics part.
If we take the equation 
$$ p_{i,j,k+1} = p_{i,j,k}(j/N^{2} + i/N) + .... $$
as the initial discrete equation here, then if we take the limit as $N \rightarrow \infty$ such that $i/N \rightarrow x$ and $j/N \rightarrow y$ we are going to find that $i/N^2 \rightarrow 0$ and $j/N^2 \rightarrow 0$. Consequently those (inverse) square terms drop out of the "continuum" version and we are left with just the first order terms and not any second order terms. (Incidentally when I do this approximation to the equation above I dont quite get the first term in the Question, but a similar first order expression, but I am not sure which of these variants is really significant, for reasons I shall discuss next.)
However the reason you give for wanting a second order equation in the first place is to model the movement of pollutants. Now these are modelled by equations like the Convection-Diffusion Equation. Here is a simplification:
$$ \frac{\partial c}{\partial t} = D \nabla ^2 c - v \cdot \nabla c $$
This equation is first order in time and has two components on the RHS:
$D \nabla ^2 c$ deriving from the Diffusion current (D is diffusion constant) and is second order.
$ v \cdot \nabla c $ deriving from the Advection current and is first order.
So it is the diffusion process which introduces second order derivatives into this equation. This in turn is because of Ficks first Law, and you might want to study the derivation of that expression at the end of the WP article, since it is derived in one dimension from movement in a grid $\Delta x$ space units wide and in time units $\Delta t$.
