# Integral of a differential equation

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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

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

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

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

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

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

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,

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

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

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

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

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

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 \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)
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=55699
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article