View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671001.png" /> in question is not linear, that is, its left-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671002.png" /> 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

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with an arbitrary function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671004.png" />; here a linear ordinary first-order differential equation corresponds to the special case

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A non-linear partial first-order differential equation for an unknown function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671006.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671007.png" /> independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671008.png" /> has the form

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n06710010.png" /> is an arbitrary function of its arguments; when

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such an equation is called quasi-linear, and when

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