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A differential equation (ordinary or partial) in which at least one of the derivatives of the unknown function (including the derivative of order zero: the function itself) occurs non-linearly. This term is used, as a rule, when one wishes to emphasize especially that the equation  $  H = 0 $
in question is not linear, that is, its left-hand side  $  H $
is not a [[Linear form|linear form]] in the derivatives of the unknown function with coefficients depending only on the independent variables.

Sometimes by a non-linear differential equation one means a more general equation of a certain form. For example, a non-linear ordinary first-order differential equation is an equation

$$ 
f \left (
x , y , 
\frac{dy}{dx}

\right )  =  0
$$

with an arbitrary function  $  f ( x , y , u ) $; 
here a linear ordinary first-order differential equation corresponds to the special case

$$ 
f ( x , y , u )  = \ 
a ( x) u + b ( x) y + c( x).
$$

A non-linear partial first-order differential equation for an unknown function  $  z $
in  $  n $
independent variables  $  x _ {1} \dots x _ {n} $
has the form

$$ 
F \left ( x _ {1} \dots x _ {n} ,\ 
z , 
\frac{\partial  z }{\partial  x _ {1} }
 \dots

\frac{\partial  z }{\partial  x _ {n} }
 \right )
 =  0 ,
$$

where  $  F $
is an arbitrary function of its arguments; when

$$ 
F  =  \sum _ { i= 1} ^ { n } 
A _ {i} ( x _ {1} \dots x _ {n} , z )

\frac{\partial  z }{\partial  x _ {i} }
 +
B ( x _ {1} \dots x _ {n} , z ) ,
$$

such an equation is called quasi-linear, and when

$$ 
F  =  \sum _ { i= 1} ^ { n } 
A _ {i} ( x _ {1} \dots x _ {n} ) 
\frac{\partial  z }{\partial  x _ {i} }
 +
$$

$$ 
+ 
B ( x _ {1} \dots x _ {n} ) z + C ( x _ {1} \dots x _ {n} ) ,
$$

it is called linear (cf. also [[Linear partial differential equation|Linear partial differential equation]]; [[Non-linear partial differential equation|Non-linear partial differential equation]]).