Non-linear differential equation

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 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; Non-linear partial differential equation).