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Difference between revisions of "Non-linear differential equation"

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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 <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.
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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
 
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671003.png" /></td> </tr></table>
+
$$
 +
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
 +
 
 +
$$
 +
f ( x , y , u )  = \
 +
a ( x) u + b ( x) y + c( x).
 +
$$
  
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
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671005.png" /></td> </tr></table>
+
$$
 +
F \left ( x _ {1} \dots x _ {n} ,\
 +
z ,
 +
\frac{\partial  z }{\partial  x _ {1} }
 +
\dots
  
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
+
\frac{\partial z }{\partial  x _ {n} }
 +
\right )
 +
= 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n0671009.png" /></td> </tr></table>
+
where  $  F $
 +
is an arbitrary function of its arguments; when
  
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
+
$$
 +
= \sum _ { i= } 1 ^ { n }
 +
A _ {i} ( x _ {1} \dots x _ {n} , z )
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n06710011.png" /></td> </tr></table>
+
\frac{\partial  z }{\partial  x _ {i} }
 +
+
 +
B ( x _ {1} \dots x _ {n} , z ) ,
 +
$$
  
 
such an equation is called quasi-linear, and when
 
such an equation is called quasi-linear, and when
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n06710012.png" /></td> </tr></table>
+
$$
 +
= \sum _ { i= } 1 ^ { n }
 +
A _ {i} ( x _ {1} \dots x _ {n} )
 +
\frac{\partial  z }{\partial  x _ {i} }
 +
+
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067100/n06710013.png" /></td> </tr></table>
+
$$
 +
+
 +
B ( x _ {1} \dots x _ {n} ) z + C ( x _ {1} \dots x _ {n} ) ,
 +
$$
  
 
it is called linear (cf. also [[Linear partial differential equation|Linear partial differential equation]]; [[Non-linear partial differential equation|Non-linear partial differential equation]]).
 
it is called linear (cf. also [[Linear partial differential equation|Linear partial differential equation]]; [[Non-linear partial differential equation|Non-linear partial differential equation]]).

Revision as of 08:03, 6 June 2020


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).

How to Cite This Entry:
Non-linear differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_differential_equation&oldid=47992
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article