This is a pattern I have noticed in some circumstances. A homogeneous function (in the sense of Euler's theorem) which is a polynomial is one in which the argument appears in the same power in each term. In my experience, multiple derivatives often follow the same pattern as binomial (or multinomial) expansion.
For example, consider the transformation to an inertial rectangular Cartesian coordinate system ($x^i$) from a rectangular Cartesian coordinate system ($x^\bar{i}$) which is rotating with angular velocity about a moving origin $\bar{\mathcal{O}}$. Notice that each term on both sides of the equations has the same number of over-dots. Notice also the similarity between the power (square) of a binomial and and the derivative of the centered part of the transformation formula for a moving point with coordinates $r^{\bar{i}}.$
$$\begin{aligned} x^{i}=&x^{i}\left(x^{\bar{j}}\right)+x_{\bar{\mathcal{O}}}^{i}\\ e_{\bar{j}}^{i}=&\frac{\partial x^{i}}{\partial x^{\bar{j}}}\\ \delta_{ij}e_{\bar{i}}^{i}e_{\bar{j}}^{j}=&\delta_{\bar{i}\bar{j}}\\ r^{i}=&e_{\bar{j}}^{i}r^{\bar{j}}+x_{\bar{\mathcal{O}}}^{i}\\ \dot{r}^{i}=&\dot{e}_{\bar{j}}^{i}r^{\bar{j}}+e_{\bar{j}}^{i}\dot{r}^{\bar{j}}+\dot{x}_{\bar{\mathcal{O}}}^{i}\\ \ddot{r}^{i}=&\ddot{e}_{\bar{j}}^{i}r^{\bar{j}}+2\dot{e}_{\bar{j}}^{i}\dot{r}^{\bar{j}}+e_{\bar{j}}^{i}\ddot{r}^{\bar{j}}+\ddot{x}_{\bar{\mathcal{O}}}^{i}\\ \left(ta+tb\right)^{2}=&t^{2}a^{2}+2t^{2}ab+t^{2}b^{2} \end{aligned} $$
The, admittedly limited, taxonomy I have for differential equations is as follows:
- type: ordinary or partial
- order: highest order derivative
- degree: highest power of the highest order derivative after the equation has been cleared of radicals and fractions in the dependent variable and its derivatives.
Notice that this does not address the number of times a lower order derivative appears.
To my mind, the facts that the number of over-dots (time derivatives) is the same in all terms, and the multinomial coefficients appear in the derivative of the centered part seem noteworthy. This also seems related to homogeneous functions.
Is this a useful topic in the treatment of differential equations?