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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
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<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>
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The graph of a solution  $  y = y ( x) $
 +
of a normal system
 +
 
 +
$$
 +
y  ^  \prime  = f ( x , y ) ,\  y \in \mathbf R  ^ {n} ,
 +
$$
  
 
of ordinary differential equations. For example, the integral curves of the equation
 
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>
+
$$
 +
y  ^  \prime  = -  
 +
\frac{x}{y}
 +
 
 +
$$
  
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
+
are the circles $  x  ^ {2} + y  ^ {2} = c  ^ {2} $,  
 +
where $  c $
 +
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>
+
$$ \tag{* }
 +
y  ^  \prime  = f ( x , y )
 +
$$
  
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.
+
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 $  ( x , y ) $
 +
the tangent of the angle of inclination of the vector with the $  x $-
 +
axis is equal to $  f( x , y ) $.  
 +
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 $  f ( x , y ) $
 +
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====
 
====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>
 
<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====
 
====Comments====
 
A normal system of differential equations is a system of differential equations of the form
 
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>
+
$$
 +
 
 +
\frac{d ^ {n _ {k} } x _ {k} }{d t ^ {n _ {k} } }
 +
=
 +
$$
  
<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>
+
$$
 +
= \
 +
F _ {k} \left ( x _ {1} ,
 +
\frac{d x _ {1} }{d t
 +
}
 +
,\dots; x _ {2} ,
 +
\frac{d x _ {2} }{d t }
 +
,\dots; \dots ;
 +
x _ {m} ,
 +
\frac{d x _ {m} }{d t }
 +
,\dots \right ) ,
 +
$$
  
<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" />.
+
$  k = 1 \dots m $,  
 +
such that the function $  F _ {k} $
 +
only depends on the $  d  ^ {j} x _ {i} / d t  ^ {j} $
 +
for $  j < n _ {i} $,  
 +
$  i= 1 \dots m $.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 22:12, 5 June 2020


The graph of a solution $ y = y ( x) $ of a normal system

$$ y ^ \prime = f ( x , y ) ,\ y \in \mathbf R ^ {n} , $$

of ordinary differential equations. For example, the integral curves of the equation

$$ y ^ \prime = - \frac{x}{y} $$

are the circles $ x ^ {2} + y ^ {2} = c ^ {2} $, where $ c $ is an arbitrary constant. The integral curve is often identified with the solution. The geometric meaning of the integral curves of a scalar equation

$$ \tag{* } y ^ \prime = f ( x , y ) $$

is the following. The equation (*) defines a direction field on the plane, that is, a field of direction vectors such that at each point $ ( x , y ) $ the tangent of the angle of inclination of the vector with the $ x $- axis is equal to $ f( x , y ) $. 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 $ f ( x , y ) $ satisfies conditions ensuring the existence and uniqueness of the Cauchy problem; the curves nowhere intersect one another and are nowhere tangent to one another.

References

[1] I.G. Petrovskii, "Ordinary differential equations" , Prentice-Hall (1966) (Translated from Russian)

Comments

A normal system of differential equations is a system of differential equations of the form

$$ \frac{d ^ {n _ {k} } x _ {k} }{d t ^ {n _ {k} } } = $$

$$ = \ F _ {k} \left ( x _ {1} , \frac{d x _ {1} }{d t } ,\dots; x _ {2} , \frac{d x _ {2} }{d t } ,\dots; \dots ; x _ {m} , \frac{d x _ {m} }{d t } ,\dots \right ) , $$

$ k = 1 \dots m $, such that the function $ F _ {k} $ only depends on the $ d ^ {j} x _ {i} / d t ^ {j} $ for $ j < n _ {i} $, $ i= 1 \dots m $.

References

[a1] G. Birkhoff, G.-C. Rota, "Ordinary differential equations" , Ginn (1962) pp. Chapt. V §5
[a2] E.L. Ince, "Ordinary differential equations" , Dover, reprint (1956) pp. §§3.6, 3.51, 4.7, A.5
How to Cite This Entry:
Integral curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_curve&oldid=47368
This article was adapted from an original article by N.N. Ladis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article