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An ordinary differential equation
 
An ordinary differential 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/d/d032/d032050/d0320501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
F ( x , y , y  ^  \prime  \dots y ^ {( n) } )  = 0
 +
$$
  
 
whose left-hand side is a total derivative:
 
whose left-hand side is a total derivative:
  
<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/d/d032/d032050/d0320502.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{d}{dx}
 +
\Phi ( x , y , y  ^  \prime  \dots y ^ {( n - 1 ) }
 +
= 0 .
 +
$$
  
In other words, equation (1) is a differential equation with total differential if there exists a differentiable function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320503.png" /> such that
+
In other words, equation (1) is a differential equation with total differential if there exists a differentiable function $  \Phi ( x , u _ {0} \dots  u _ {n - 1 }  ) $
 +
such that
  
<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/d/d032/d032050/d0320504.png" /></td> </tr></table>
+
$$
 +
F ( x , u _ {0} \dots u _ {n} )  \equiv  \Phi _ {x}  ^  \prime
 +
+ u _ {1} \Phi _ {u _ {0}  }  ^  \prime  +
 +
+ \dots + u _ {n} \Phi _ {u _ {n-} 1 }  ^  \prime
 +
$$
  
identically with respect to all arguments. The solution of a differential equation with total differential of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320505.png" /> is reduced to solving an equation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320506.png" />:
+
identically with respect to all arguments. The solution of a differential equation with total differential of order $  n $
 +
is reduced to solving an equation of order $  ( n - 1 ) $:
  
<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/d/d032/d032050/d0320507.png" /></td> </tr></table>
+
$$
 +
\Phi ( x , y , y  ^  \prime  \dots y ^ {( n - 1 ) } )  = C ,\ \
 +
C = \textrm{ const } .
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320508.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d0320509.png" /> times continuously-differentiable function and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205010.png" /> be a function having continuous partial derivatives up to and including the second order. Let
+
Let $  F ( x, u _ {0} \dots u _ {n} ) $
 +
be an $  n $
 +
times continuously-differentiable function and let $  \Phi ( x , u _ {0} \dots  u _ {n - 1 }  ) $
 +
be a function having continuous partial derivatives up to and including the second order. Let
  
<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/d/d032/d032050/d03205011.png" /></td> </tr></table>
+
$$
 +
\Delta \Phi  = \Phi _ {x}  ^  \prime  + u _ {1} \Phi _ {u _ {0}  }  ^  \prime
 +
+ \dots + u _ {n} \Phi _ {u _ {n-} 1 }  ^  \prime  ,
 +
$$
  
<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/d/d032/d032050/d03205012.png" /></td> </tr></table>
+
$$
 +
\Delta _ {0} F  = F _ {u _ {n}  } ^ { \prime } ,\  \Delta _  \nu  F  = F _ {u _ {n - \nu }  } ^ { \prime } - \Delta (
 +
\Delta _ {\nu - 1 }  F ) ,\  \nu = 1 \dots n .
 +
$$
  
For equation (1) to be a differential equation with total differential it is sufficient that the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205014.png" />, are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205015.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205016.png" /> [[#References|[1]]]. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205017.png" /> may enter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205018.png" /> in a linear manner only.
+
For equation (1) to be a differential equation with total differential it is sufficient that the functions $  \Delta _  \nu  F $,  
 +
$  \nu = 0 \dots n $,  
 +
are independent of $  u _ {n} $
 +
and that $  \Delta _ {n }  F = 0 $[[#References|[1]]]. In particular, $  u _ {n} $
 +
may enter $  F $
 +
in a linear manner only.
  
 
The first-order equation
 
The first-order 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/d/d032/d032050/d03205019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
M ( x , y ) + N ( x , y ) y  ^  \prime  = 0 ,
 +
$$
  
where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205023.png" /> are defined and continuous in an open simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205024.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205025.png" />-plane and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205027.png" />, is a differential equation with total differential if and only if
+
where the functions $  M $,  
 +
$  N $,  
 +
$  M _ {y} ^ { \prime } $,  
 +
and $  N _ {x} ^ { \prime } $
 +
are defined and continuous in an open simply-connected domain $  D $
 +
of the $  ( x , y ) $-
 +
plane and $  M  ^ {2} + N  ^ {2} > 0 $
 +
in $  D $,  
 +
is a differential equation with total differential if and only if
  
<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/d/d032/d032050/d03205028.png" /></td> </tr></table>
+
$$
 +
M _ {y} ^ { \prime } ( x , y )  \equiv  N _ {x} ^ { \prime }
 +
( x , y ) \  \mathop{\rm in}  D .
 +
$$
  
The general solution of equation (2) with total differential has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205029.png" />, where
+
The general solution of equation (2) with total differential has the form $  \Phi ( x , y ) = 0 $,  
 +
where
  
<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/d/d032/d032050/d03205030.png" /></td> </tr></table>
+
$$
 +
\Phi ( x , y )  = \int\limits _ {( x _ {0} , y _ {0} ) } ^ { {( }  x , y ) } M ( x , y )  dx + N ( x , y )  dy ,
 +
$$
  
and the integral is taken over any rectifiable curve lying inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205031.png" /> and joining an arbitrary fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205032.png" /> with the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205033.png" /> [[#References|[2]]]. Equation (2) (in the general case, an equation (1) which is linear with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032050/d03205034.png" />) can, under certain conditions, be reduced to a differential equation with total differential by multiplying by an [[Integrating factor|integrating factor]].
+
and the integral is taken over any rectifiable curve lying inside $  D $
 +
and joining an arbitrary fixed point $  ( x _ {0} , y _ {0} ) \in D $
 +
with the point $  ( x , y ) $[[#References|[2]]]. Equation (2) (in the general case, an equation (1) which is linear with respect to $  y  ^ {(} n) $)  
 +
can, under certain conditions, be reduced to a differential equation with total differential by multiplying by an [[Integrating factor|integrating factor]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Erugin,  "A general course in differential equations" , Minsk  (1972)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kamke,  "Differentialgleichungen: Lösungen und Lösungsmethoden" , '''1. Gewöhnliche Differentialgleichungen''' , Chelsea, reprint  (1947)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.P. Erugin,  "A general course in differential equations" , Minsk  (1972)  (In Russian)</TD></TR></table>

Latest revision as of 17:33, 5 June 2020


An ordinary differential equation

$$ \tag{1 } F ( x , y , y ^ \prime \dots y ^ {( n) } ) = 0 $$

whose left-hand side is a total derivative:

$$ \frac{d}{dx} \Phi ( x , y , y ^ \prime \dots y ^ {( n - 1 ) } ) = 0 . $$

In other words, equation (1) is a differential equation with total differential if there exists a differentiable function $ \Phi ( x , u _ {0} \dots u _ {n - 1 } ) $ such that

$$ F ( x , u _ {0} \dots u _ {n} ) \equiv \Phi _ {x} ^ \prime + u _ {1} \Phi _ {u _ {0} } ^ \prime + + \dots + u _ {n} \Phi _ {u _ {n-} 1 } ^ \prime $$

identically with respect to all arguments. The solution of a differential equation with total differential of order $ n $ is reduced to solving an equation of order $ ( n - 1 ) $:

$$ \Phi ( x , y , y ^ \prime \dots y ^ {( n - 1 ) } ) = C ,\ \ C = \textrm{ const } . $$

Let $ F ( x, u _ {0} \dots u _ {n} ) $ be an $ n $ times continuously-differentiable function and let $ \Phi ( x , u _ {0} \dots u _ {n - 1 } ) $ be a function having continuous partial derivatives up to and including the second order. Let

$$ \Delta \Phi = \Phi _ {x} ^ \prime + u _ {1} \Phi _ {u _ {0} } ^ \prime + \dots + u _ {n} \Phi _ {u _ {n-} 1 } ^ \prime , $$

$$ \Delta _ {0} F = F _ {u _ {n} } ^ { \prime } ,\ \Delta _ \nu F = F _ {u _ {n - \nu } } ^ { \prime } - \Delta ( \Delta _ {\nu - 1 } F ) ,\ \nu = 1 \dots n . $$

For equation (1) to be a differential equation with total differential it is sufficient that the functions $ \Delta _ \nu F $, $ \nu = 0 \dots n $, are independent of $ u _ {n} $ and that $ \Delta _ {n } F = 0 $[1]. In particular, $ u _ {n} $ may enter $ F $ in a linear manner only.

The first-order equation

$$ \tag{2 } M ( x , y ) + N ( x , y ) y ^ \prime = 0 , $$

where the functions $ M $, $ N $, $ M _ {y} ^ { \prime } $, and $ N _ {x} ^ { \prime } $ are defined and continuous in an open simply-connected domain $ D $ of the $ ( x , y ) $- plane and $ M ^ {2} + N ^ {2} > 0 $ in $ D $, is a differential equation with total differential if and only if

$$ M _ {y} ^ { \prime } ( x , y ) \equiv N _ {x} ^ { \prime } ( x , y ) \ \mathop{\rm in} D . $$

The general solution of equation (2) with total differential has the form $ \Phi ( x , y ) = 0 $, where

$$ \Phi ( x , y ) = \int\limits _ {( x _ {0} , y _ {0} ) } ^ { {( } x , y ) } M ( x , y ) dx + N ( x , y ) dy , $$

and the integral is taken over any rectifiable curve lying inside $ D $ and joining an arbitrary fixed point $ ( x _ {0} , y _ {0} ) \in D $ with the point $ ( x , y ) $[2]. Equation (2) (in the general case, an equation (1) which is linear with respect to $ y ^ {(} n) $) can, under certain conditions, be reduced to a differential equation with total differential by multiplying by an integrating factor.

References

[1] E. Kamke, "Differentialgleichungen: Lösungen und Lösungsmethoden" , 1. Gewöhnliche Differentialgleichungen , Chelsea, reprint (1947)
[2] N.P. Erugin, "A general course in differential equations" , Minsk (1972) (In Russian)
How to Cite This Entry:
Differential equation with total differential. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Differential_equation_with_total_differential&oldid=14295
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article