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 in question is not linear, that is, its left-hand side 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

with an arbitrary function ; here a linear ordinary first-order differential equation corresponds to the special case

A non-linear partial first-order differential equation for an unknown function in independent variables has the form

where is an arbitrary function of its arguments; when

such an equation is called quasi-linear, and when

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