View source for Integral curve

The graph of a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i0513801.png" /> of a normal system

<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/i051380/i0513802.png" /></td> </tr></table>

of ordinary differential equations. For example, the integral curves of 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/i051380/i0513803.png" /></td> </tr></table>

are the circles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i0513804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i0513805.png" /> is an arbitrary constant. The integral curve is often identified with the solution. The geometric meaning of the integral curves of a scalar 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/i051380/i0513806.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

is the following. The equation (*) defines a [[Direction field|direction field]] on the plane, that is, a field of direction vectors such that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i0513807.png" /> the tangent of the angle of inclination of the vector with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i0513808.png" />-axis is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i0513809.png" />. The integral curves of (*) are then the curves that at each point have a tangent coinciding with the vector of the direction field at this point. The integral curves of (*) fill out the entire region in which the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i05138010.png" /> satisfies conditions ensuring the existence and uniqueness of the [[Cauchy problem|Cauchy problem]]; the curves nowhere intersect one another and are nowhere tangent to one another.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.G. Petrovskii,   "Ordinary differential equations" , Prentice-Hall  (1966)  (Translated from Russian)</TD></TR></table>



====Comments====
A normal system of differential equations is a system of differential equations 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/i051380/i05138011.png" /></td> </tr></table>

<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/i051380/i05138012.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i05138013.png" />, such that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i05138014.png" /> only depends on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i05138015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i05138016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051380/i05138017.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Birkhoff,   G.-C. Rota,   "Ordinary differential equations" , Ginn  (1962)  pp. Chapt. V §5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Ince,   "Ordinary differential equations" , Dover, reprint  (1956)  pp. §§3.6, 3.51, 4.7, A.5</TD></TR></table>