Linear partial differential equation
From HandWiki
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.
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