View source for 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515501.png" /> defining a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515502.png" /> of 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/i/i051/i051550/i0515503.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

as an implicit function of the independent variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515504.png" />. 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

<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/i/i051/i051550/i0515505.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

from which one can obtain by an appropriate choice of the constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515506.png" /> any [[Integral curve|integral curve]] of (1) lying in some given region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515507.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515508.png" />-plane. If the arbitrary constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i0515509.png" /> are eliminated from equation (2) and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155010.png" /> relations obtained from it by repeated differentiation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155011.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155012.png" /> is regarded as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155013.png" />), then equation (1) results. A relation of the form

<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/i/i051/i051550/i05155014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>

containing derivatives up to order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155016.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155017.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155018.png" /> is reduced to the solution of equation (3) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155019.png" />. If (3) contains just one arbitrary constant, that is, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155020.png" />, then it is called a first integral of (1). This equation has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155021.png" /> independent first integrals; knowledge of such integrals enables one to obtain the general solution of (1) by eliminating the quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155022.png" /> from them.

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

<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/i/i051/i051550/i05155023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>

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

<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/i/i051/i051550/i05155024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155025.png" /> are arbitrary constants, which describes in implicit form all the solutions of the system (4) in some region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155026.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155027.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155028.png" /> with the property that it is constant along any solution of the system (4) in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155029.png" />. The system (4) has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155030.png" /> independent first integrals, knowledge of which enables one to find the general solution without integrating the system; knowledge of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155031.png" /> independent first integrals enables one to reduce the solution of the system (4) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155032.png" /> to the solution of a system of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155033.png" />. A smooth function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155034.png" /> is a first integral of the system (4) with smooth right-hand side if and only if it satisfies the 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/i/i051/i051550/i05155035.png" /></td> </tr></table>

Similar terminology is sometimes used in the theory of first-order partial differential equations. Thus, by an integral of the 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/i/i051/i051550/i05155036.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051550/i05155037.png" /> 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====
<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>



====Comments====


====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>