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 \geq 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.