Linear partial differential equation

An equation of the form

$$F(x\dots p_{i_1\dots i_n},...)=0,$$

where $F$ is a linear function of real variables,

$$p_{i_1\dots i_n}\equiv \frac{{\partial }^k}{\partial x_1^{i_1}\dots d{x}_n^{i_n}},$$

$i_1\dots i_n$ are non-negative integer indices, $\sum _{j=1}^n{i}_j=k$, $k=0\dots m$, $m\ge 1$, and at least one of the derivatives

$$\frac{\partial F}{\partial p_{i_1\dots i_n}},\sum _{j=1}^n{i}_j=m,$$

is non-zero.

For more details, see Differential equation, partial.