Integral of a differential equation

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

[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)