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Difference between revisions of "Integral of a differential equation"

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m (eqref)
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of an ordinary differential equation
 
of an ordinary differential equation
  
$$ \tag{1 }
+
\begin{equation}\label{eq1} \tag{1 }
F ( x , y , y  ^  \prime  \dots y  ^ {(} n) )  =  0
+
F ( x , y , y  ^  \prime  \dots y  ^ {(n)} )  =  0
$$
+
\end{equation}
  
 
as an implicit function of the independent variable  $  x $.  
 
as an implicit function of the independent variable  $  x $.  
The solution is in this case also said to be a particular integral, in contrast to the general integral of equation (1), that is, a relation
+
The solution is in this case also said to be a particular integral, in contrast to the general integral of equation \eqref{eq1}, that is, a relation
  
$$ \tag{2 }
+
\begin{equation}\label{eq2} \tag{2 }
 
\Phi ( x , y , C _ {1} \dots C _ {n} )  =  0 ,
 
\Phi ( x , y , C _ {1} \dots C _ {n} )  =  0 ,
$$
+
\end{equation}
  
 
from which one can obtain by an appropriate choice of the constants  $  C _ {1} \dots C _ {n} $
 
from which one can obtain by an appropriate choice of the constants  $  C _ {1} \dots C _ {n} $
any [[Integral curve|integral curve]] of (1) lying in some given region  $  G $
+
any [[integral curve]] of \eqref{eq1} lying in some given region  $  G $
 
of the  $  ( x , y ) $-
 
of the  $  ( x , y ) $-
 
plane. If the arbitrary constants  $  C _ {1} \dots C _ {n} $
 
plane. If the arbitrary constants  $  C _ {1} \dots C _ {n} $
are eliminated from equation (2) and the $ n $
+
are eliminated from equation \eqref{eq2} and the $n$ relations obtained from it by repeated differentiation with respect to $x$ (
relations obtained from it by repeated differentiation with respect to $ x $(
 
 
where  $  y $
 
where  $  y $
 
is regarded as a function of  $  x $),  
 
is regarded as a function of  $  x $),  
then equation (1) results. A relation of the form
+
then equation \eqref{eq1} results. A relation of the form
  
$$ \tag{3 }
+
\begin{equation}\label{eq3} \tag{3 }
\Phi ( x , y , y  ^  \prime  \dots y  ^ {(} k) ,\  
+
\Phi ( x , y , y  ^  \prime  \dots y  ^ {(k)} ,\  
 
C _ {1} \dots C _ {n-} k )  =  0 ,
 
C _ {1} \dots C _ {n-} k )  =  0 ,
$$
+
\end{equation}
  
 
containing derivatives up to order  $  k $,  
 
containing derivatives up to order  $  k $,  
 
$  1 \leq  k < n $,  
 
$  1 \leq  k < n $,  
 
and  $  n - k $
 
and  $  n - k $
arbitrary constants, arising from the process of integrating equation (1), is sometimes called an intermediate integral of equation (1). If an intermediate integral (3) is known, then the solution of equation (1) of order  $  n $
+
arbitrary constants, arising from the process of integrating equation \eqref{eq1}, is sometimes called an intermediate integral of equation \eqref{eq1}. If an intermediate integral \eqref{eq3} is known, then the solution of equation \eqref{eq1} of order  $  n $
is reduced to the solution of equation (3) of order  $  k $.  
+
is reduced to the solution of equation \eqref{eq3} of order  $  k $.  
If (3) contains just one arbitrary constant, that is, if  $  k = n - 1 $,  
+
If \eqref{eq3} contains just one arbitrary constant, that is, if  $  k = n - 1 $,  
then it is called a first integral of (1). This equation has exactly  $  n $
+
then it is called a first integral of \eqref{eq1}. This equation has exactly  $  n $
independent first integrals; knowledge of such integrals enables one to obtain the general solution of (1) by eliminating the quantities  $  y  ^  \prime  \dots y  ^ {(} n- 1) $
+
independent first integrals; knowledge of such integrals enables one to obtain the general solution of (1) by eliminating the quantities  $  y  ^  \prime  \dots y  ^ {( n- 1)} $
 
from them.
 
from them.
  
 
If one considers a first-order system of ordinary differential equations,
 
If one considers a first-order system of ordinary differential equations,
  
$$ \tag{4 }
+
\begin{equation}\label{eq4} \tag{4 }
  
 
\frac{d x _ {i} }{dt}
 
\frac{d x _ {i} }{dt}
Line 59: Line 58:
 
f _ {i} ( t , x _ {1} \dots x _ {n} ) ,\ \  
 
f _ {i} ( t , x _ {1} \dots x _ {n} ) ,\ \  
 
i = 1 \dots n ,
 
i = 1 \dots n ,
$$
+
\end{equation}
 +
 
  
 
then by a general integral of it is meant a set of relations
 
then by a general integral of it is meant a set of relations
  
$$ \tag{5 }
+
\begin{equation}\label{eq5} \tag{5 }
 
\Phi _ {i} ( t , x _ {1} \dots x _ {n} )  =  C _ {i} ,\ \  
 
\Phi _ {i} ( t , x _ {1} \dots x _ {n} )  =  C _ {i} ,\ \  
 
i = 1 \dots n ,
 
i = 1 \dots n ,
$$
+
\end{equation}
  
 
where the  $  C _ {i} $
 
where the  $  C _ {i} $
are arbitrary constants, which describes in implicit form all the solutions of the system (4) in some region  $  G $
+
are arbitrary constants, which describes in implicit form all the solutions of the system \eqref{eq4} in some region  $  G $
 
of the  $  ( t , x _ {1} \dots x _ {n} ) $-
 
of the  $  ( t , x _ {1} \dots x _ {n} ) $-
space. Each of the relations (5) is itself called a first integral of the system (4). More often, by a first integral of the system (4) one means a function  $  u ( t , x _ {1} \dots x _ {n} ) $
+
space. Each of the relations \eqref{eq5} is itself called a first integral of the system \eqref{eq4}. More often, by a first integral of the system \eqref{eq4} one means a function  $  u ( t , x _ {1} \dots x _ {n} ) $
with the property that it is constant along any solution of the system (4) in a region  $  G $.  
+
with the property that it is constant along any solution of the system \eqref{eq4} in a region  $  G $.  
The system (4) has exactly  $  n $
+
The system \eqref{eq4} has exactly  $  n $
 
independent first integrals, knowledge of which enables one to find the general solution without integrating the system; knowledge of  $  k $
 
independent first integrals, knowledge of which enables one to find the general solution without integrating the system; knowledge of  $  k $
independent first integrals enables one to reduce the solution of the system (4) of order  $  n $
+
independent first integrals enables one to reduce the solution of the system \eqref{eq4} of order  $  n $
 
to the solution of a system of order  $  n - k $.  
 
to the solution of a system of order  $  n - k $.  
 
A smooth function  $  u ( t , x _ {1} \dots x _ {n} ) $
 
A smooth function  $  u ( t , x _ {1} \dots x _ {n} ) $
Line 84: Line 84:
 
\frac{\partial  u }{\partial  t }
 
\frac{\partial  u }{\partial  t }
 
  +
 
  +
\sum _ { i= } 1 ^ { n }  
+
\sum _ { i=1}^ { n }  
 
f _ {i} ( t , x _ {1} \dots x _ {n} )
 
f _ {i} ( t , x _ {1} \dots x _ {n} )
  
Line 93: Line 93:
 
Similar terminology is sometimes used in the theory of first-order partial differential equations. Thus, by an integral of the differential equation
 
Similar terminology is sometimes used in the theory of first-order partial differential equations. Thus, by an integral of the differential equation
  
$$ \tag{6 }
+
\begin{equation}\label{eq6} \tag{6 }
 
F \left ( x , y , z ,\  
 
F \left ( x , y , z ,\  
  
Line 102: Line 102:
  
 
\right )  =  0 ,
 
\right )  =  0 ,
$$
+
\end{equation}
  
or by a particular integral of it, is meant a solution of this equation (an [[Integral surface|integral surface]]). By a complete integral of (6) is meant a family of solutions  $  \Phi ( x , y , z , a , b ) = 0 $
+
or by a particular integral of it, is meant a solution of this equation (an [[integral surface]]). By a complete integral of \eqref{eq6} is meant a family of solutions  $  \Phi ( x , y , z , a , b ) = 0 $
 
depending on two arbitrary constants. A general integral of equation (6) is a relation containing one arbitrary function and giving a solution of the equation for each choice of this function.
 
depending on two arbitrary constants. A general integral of equation (6) is a relation containing one arbitrary function and giving a solution of the equation for each choice of this function.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,  "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  W.W. [V.V. Stepanov] Stepanow,  "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR>
====Comments====
+
<TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Survey of applicable mathematics'' , Iliffe  (1969)  pp. Sects. 17.2, 17.8, 17.18, 17.20</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Ince,  "Integration of ordinary differential equations" , Oliver &amp; Boyd  (1956)</TD></TR>
 
+
</table>
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Rektorys (ed.) , ''Survey of applicable mathematics'' , Iliffe  (1969)  pp. Sects. 17.2, 17.8, 17.18, 17.20</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Ince,  "Integration of ordinary differential equations" , Oliver &amp; Boyd  (1956)</TD></TR></table>
 

Revision as of 12:06, 31 March 2024


A solution of the differential equation. By an integral of a differential equation is primarily meant a relation of the form $ \Phi ( x , y ) = 0 $ defining a solution $ y $ of an ordinary differential equation

\begin{equation}\label{eq1} \tag{1 } F ( x , y , y ^ \prime \dots y ^ {(n)} ) = 0 \end{equation}

as an implicit function of the independent variable $ x $. The solution is in this case also said to be a particular integral, in contrast to the general integral of equation \eqref{eq1}, that is, a relation

\begin{equation}\label{eq2} \tag{2 } \Phi ( x , y , C _ {1} \dots C _ {n} ) = 0 , \end{equation}

from which one can obtain by an appropriate choice of the constants $ C _ {1} \dots C _ {n} $ any integral curve of \eqref{eq1} lying in some given region $ G $ of the $ ( x , y ) $- plane. If the arbitrary constants $ C _ {1} \dots C _ {n} $ are eliminated from equation \eqref{eq2} and the $n$ relations obtained from it by repeated differentiation with respect to $x$ ( where $ y $ is regarded as a function of $ x $), then equation \eqref{eq1} results. A relation of the form

\begin{equation}\label{eq3} \tag{3 } \Phi ( x , y , y ^ \prime \dots y ^ {(k)} ,\ C _ {1} \dots C _ {n-} k ) = 0 , \end{equation}

containing derivatives up to order $ k $, $ 1 \leq k < n $, and $ n - k $ arbitrary constants, arising from the process of integrating equation \eqref{eq1}, is sometimes called an intermediate integral of equation \eqref{eq1}. If an intermediate integral \eqref{eq3} is known, then the solution of equation \eqref{eq1} of order $ n $ is reduced to the solution of equation \eqref{eq3} of order $ k $. If \eqref{eq3} contains just one arbitrary constant, that is, if $ k = n - 1 $, then it is called a first integral of \eqref{eq1}. This equation has exactly $ n $ independent first integrals; knowledge of such integrals enables one to obtain the general solution of (1) by eliminating the quantities $ y ^ \prime \dots y ^ {( n- 1)} $ from them.

If one considers a first-order system of ordinary differential equations,

\begin{equation}\label{eq4} \tag{4 } \frac{d x _ {i} }{dt} = \ f _ {i} ( t , x _ {1} \dots x _ {n} ) ,\ \ i = 1 \dots n , \end{equation}


then by a general integral of it is meant a set of relations

\begin{equation}\label{eq5} \tag{5 } \Phi _ {i} ( t , x _ {1} \dots x _ {n} ) = C _ {i} ,\ \ i = 1 \dots n , \end{equation}

where the $ C _ {i} $ are arbitrary constants, which describes in implicit form all the solutions of the system \eqref{eq4} in some region $ G $ of the $ ( t , x _ {1} \dots x _ {n} ) $- space. Each of the relations \eqref{eq5} is itself called a first integral of the system \eqref{eq4}. More often, by a first integral of the system \eqref{eq4} one means a function $ u ( t , x _ {1} \dots x _ {n} ) $ with the property that it is constant along any solution of the system \eqref{eq4} in a region $ G $. The system \eqref{eq4} has exactly $ n $ independent first integrals, knowledge of which enables one to find the general solution without integrating the system; knowledge of $ k $ independent first integrals enables one to reduce the solution of the system \eqref{eq4} of order $ n $ to the solution of a system of order $ n - k $. A smooth function $ u ( t , x _ {1} \dots x _ {n} ) $ is a first integral of the system (4) with smooth right-hand side if and only if it satisfies the equation

$$ \frac{\partial u }{\partial t } + \sum _ { i=1}^ { n } f _ {i} ( t , x _ {1} \dots x _ {n} ) \frac{\partial u }{\partial x _ {i} } = 0 . $$

Similar terminology is sometimes used in the theory of first-order partial differential equations. Thus, by an integral of the differential equation

\begin{equation}\label{eq6} \tag{6 } F \left ( x , y , z ,\ \frac{\partial z }{\partial x } ,\ \frac{\partial z }{\partial y } \right ) = 0 , \end{equation}

or by a particular integral of it, is meant a solution of this equation (an integral surface). By a complete integral of \eqref{eq6} is meant a family of solutions $ \Phi ( x , y , z , a , b ) = 0 $ depending on two arbitrary constants. A general integral of equation (6) is a relation containing one arbitrary function and giving a solution of the equation for each choice of this function.

References

[1] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[a1] K. Rektorys (ed.) , Survey of applicable mathematics , Iliffe (1969) pp. Sects. 17.2, 17.8, 17.18, 17.20
[a2] E.L. Ince, "Integration of ordinary differential equations" , Oliver & Boyd (1956)
How to Cite This Entry:
Integral of a differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_of_a_differential_equation&oldid=55698
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article