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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 \eqref{eq6} is a relation containing one arbitrary function and giving a solution of the equation for each choice of this function.

====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>
<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>