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''of a first-order differential equation
 
''of a first-order 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/i052/i052720/i0527201.png" /></td> </tr></table>
+
$$ \tag{* }
 +
y  ^  \prime  = f ( x, y)
 +
$$
  
 
''
 
''
  
A set of points in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i0527202.png" />-plane at which the inclinations of the [[Direction field|direction field]] defined by equation
+
A set of points in the $  ( x, y) $-
 +
plane at which the inclinations of the [[Direction field|direction field]] defined by equation
  
are one and the same. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i0527203.png" /> is an arbitrary real number, then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i0527205.png" />-isocline of equation
+
are one and the same. If $  k $
 +
is an arbitrary real number, then the $  k $-
 +
isocline of equation
  
 
is the set
 
is the set
  
<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/i052/i052720/i0527206.png" /></td> </tr></table>
+
$$
 +
\{ {( x, y) } : {f ( x, y) = k } \}
 +
$$
  
(in general, this is a curve); at each of its points the (oriented) angle between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i0527207.png" />-axis and the tangent to the solution of
+
(in general, this is a curve); at each of its points the (oriented) angle between the $  x $-
 +
axis and the tangent to the solution of
  
going through the point is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i0527208.png" />. For example, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i0527209.png" />-isocline is defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272010.png" /> and consists of just those points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272011.png" />-plane at which the solutions of equation
+
going through the point is $  { \mathop{\rm arc}  \mathop{\rm tan} }  k $.  
 +
For example, the 0 $-
 +
isocline is defined by the equation $  f ( x, y) = 0 $
 +
and consists of just those points of the $  ( x, y) $-
 +
plane at which the solutions of equation
  
have horizontal tangents. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272012.png" />-isocline of
+
have horizontal tangents. The $  k $-
 +
isocline of
  
 
is simultaneously a solution of
 
is simultaneously a solution of
  
if and only if it is a line with slope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272013.png" />.
+
if and only if it is a line with slope $  k $.
  
 
A rough qualitative representation of the behaviour of the integral curves (cf. [[Integral curve|Integral curve]]) of
 
A rough qualitative representation of the behaviour of the integral curves (cf. [[Integral curve|Integral curve]]) of
  
can be obtained if the isoclines of the given equation are constructed for a sufficiently frequent choice of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272014.png" />, and if the corresponding inclinations of the integral curves are drawn (the method of isoclines). It is also useful to construct the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272015.png" />-isocline, defined by the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272016.png" />; at the points of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272017.png" />-isocline the integral curves of equation
+
can be obtained if the isoclines of the given equation are constructed for a sufficiently frequent choice of the parameter $  k $,  
 +
and if the corresponding inclinations of the integral curves are drawn (the method of isoclines). It is also useful to construct the $  \infty $-
 +
isocline, defined by the equation $  1/ {f ( x, y) } = 0 $;  
 +
at the points of the $  \infty $-
 +
isocline the integral curves of equation
  
 
have vertical tangents. The (local) extreme points of the solutions of
 
have vertical tangents. The (local) extreme points of the solutions of
  
can lie on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272018.png" />-isocline only, and the points of inflection of the solution can lie only on the curve
+
can lie on the 0 $-
 +
isocline only, and the points of inflection of the solution can lie only on the curve
  
<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/i052/i052720/i05272019.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{\partial  f ( x, y) }{\partial  x }
 +
+
 +
f ( x, y)
 +
 
 +
\frac{\partial  f ( x, y) }{\partial  y }
 +
  = 0.
 +
$$
  
 
For a first-order equation not solvable with respect to the derivative,
 
For a first-order equation not solvable with respect to the derivative,
  
<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/i052/i052720/i05272020.png" /></td> </tr></table>
+
$$
 +
F ( x, y, y  ^  \prime  )  = 0,
 +
$$
  
the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052720/i05272021.png" />-isocline is defined as the set
+
the $  k $-
 +
isocline is defined as the set
  
<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/i052/i052720/i05272022.png" /></td> </tr></table>
+
$$
 +
\{ {( x, y) } : {F ( x, y, k) = 0 } \}
 +
.
 +
$$
  
 
In the case of a second-order autonomous system,
 
In the case of a second-order autonomous 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/i052/i052720/i05272023.png" /></td> </tr></table>
+
$$
 +
\dot{x}  = f ( x, y),\ \
 +
\dot{y}  = g ( x, y),
 +
$$
  
 
the set of points in the phase plane at which the vectors of the phase velocity are collinear is an isocline of the equation
 
the set of points in the phase plane at which the vectors of the phase velocity are collinear is an isocline 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/i052/i052720/i05272024.png" /></td> </tr></table>
+
$$
 +
 
 +
\frac{dy }{dx }
 +
  = \
 +
 
 +
\frac{g ( x, y) }{f ( x, y) }
 +
.
 +
$$
  
 
====References====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.T. Davis,  "Introduction to nonlinear differential and integral equations" , Dover, reprint  (1962)  pp. Chapt. II, §2</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.T. Davis,  "Introduction to nonlinear differential and integral equations" , Dover, reprint  (1962)  pp. Chapt. II, §2</TD></TR></table>

Latest revision as of 22:13, 5 June 2020


of a first-order differential equation

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

A set of points in the $ ( x, y) $- plane at which the inclinations of the direction field defined by equation

are one and the same. If $ k $ is an arbitrary real number, then the $ k $- isocline of equation

is the set

$$ \{ {( x, y) } : {f ( x, y) = k } \} $$

(in general, this is a curve); at each of its points the (oriented) angle between the $ x $- axis and the tangent to the solution of

going through the point is $ { \mathop{\rm arc} \mathop{\rm tan} } k $. For example, the $ 0 $- isocline is defined by the equation $ f ( x, y) = 0 $ and consists of just those points of the $ ( x, y) $- plane at which the solutions of equation

have horizontal tangents. The $ k $- isocline of

is simultaneously a solution of

if and only if it is a line with slope $ k $.

A rough qualitative representation of the behaviour of the integral curves (cf. Integral curve) of

can be obtained if the isoclines of the given equation are constructed for a sufficiently frequent choice of the parameter $ k $, and if the corresponding inclinations of the integral curves are drawn (the method of isoclines). It is also useful to construct the $ \infty $- isocline, defined by the equation $ 1/ {f ( x, y) } = 0 $; at the points of the $ \infty $- isocline the integral curves of equation

have vertical tangents. The (local) extreme points of the solutions of

can lie on the $ 0 $- isocline only, and the points of inflection of the solution can lie only on the curve

$$ \frac{\partial f ( x, y) }{\partial x } + f ( x, y) \frac{\partial f ( x, y) }{\partial y } = 0. $$

For a first-order equation not solvable with respect to the derivative,

$$ F ( x, y, y ^ \prime ) = 0, $$

the $ k $- isocline is defined as the set

$$ \{ {( x, y) } : {F ( x, y, k) = 0 } \} . $$

In the case of a second-order autonomous system,

$$ \dot{x} = f ( x, y),\ \ \dot{y} = g ( x, y), $$

the set of points in the phase plane at which the vectors of the phase velocity are collinear is an isocline of the equation

$$ \frac{dy }{dx } = \ \frac{g ( x, y) }{f ( x, y) } . $$

References

[1] W.W. [V.V. Stepanov] Stepanow, "Lehrbuch der Differentialgleichungen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)

Comments

References

[a1] H.T. Davis, "Introduction to nonlinear differential and integral equations" , Dover, reprint (1962) pp. Chapt. II, §2
How to Cite This Entry:
Isocline. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isocline&oldid=16597
This article was adapted from an original article by N.Kh. Rozov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article