View source for Linear partial differential equation

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An equation of the form

$$ 
F ( x \dots p _ {i _ {1}  \dots i _ {n} } , .  .  . )  =  0 ,
$$

where  $  F $
is a [[Linear function|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|Differential equation, partial]].