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 | + | <!-- |
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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 | ||
| − | + | $$ | |
| + | f \left ( | ||
| + | x , y , | ||
| + | \frac{dy}{dx} | ||
| + | |||
| + | \right ) = 0 | ||
| + | $$ | ||
| + | |||
| + | 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 | 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]]). | 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).
Non-linear differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Non-linear_differential_equation&oldid=14678